High School: Statistics & Probability - Common Core: High School - Statistics and Probability

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events, we can simply add the probabilities, which can be illustrated by the following figure. In this figure, each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and, at times, intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and the generation of probability distribution models. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand probabilities in a general sense, we need to determine how we can create a probability distribution graph for a probability model. We will use the following equation to calculate the probabilities to be used in this graphical display:

Remember that in combinations and permutations a combination is calculated using the following formula:

Now, we can write the following formula.

In this formula variables are defined in the following manner:

Let's investigate this standard through the use of an example. Suppose a researcher rolls a die twelve times and notes whether the die rolls on an even or an odd number. If the die is fair (i.e. every number has an equal probability of being rolled or each roll is random), then would the probability distribution graph follow the pattern of a normal distribution? Lets create a table and solve for each variable. The probability of rolling an even number—a two, four, or a six— is three out of six or fifty percent. Likewise, the probability of failure (i.e. rolling an odd number) is three out of six or fifty percent. Next, we need to list the number of successes for each event—variable . The researcher can roll an even number every time he rolls and he may not even roll an even number in all twelve trial. We need to calculate this probability for every possible number of successful events; therefore, the number of successes ranges from zero to twelve. Last, we know that there are a total of twelve trials. We have solved for the probability of each of these variables for every possible number of successes in the trials.

Once this data is tabulated we can graph the probability of rolling an even number. If we look at the graph then we can see if it follows the bell shape curve of a normal distribution.

A bell curve is shaped like the following image:

We can quickly tell that the graph of the probability distribution does follow the shape of a normal or "bell" curve.

Let's use this information to solve the question. If we look at the question, then we know that the probability of turning right or left in a random situation is fifty percent; therefore, the probability of success (i.e. the right path) or failure (i.e. the left path) is one half or fifty percent. Next we know that there were ten people questioned—or ten trials—and the number of successes per trial ranged from zero to ten. This information has been tabulated and the probability of choosing the left or right has been calculated.

We can now graph the probabilities.

We can see that the graph follows a normal distribution with a characteristic bell shape.

In order to solve this problem, let's first observe the steps associated with the scientific method. In this particular experiment we can see that the scientists have interest in a particular biological phenomenon: the effects of hormones on plant growth and germination. They develop a hypothesis based on these observations. A hypothesis is a tentative explanation for an observed phenomenon. In this experiment the scientists developed the following hypothesis: abscisic acid will inhibit plant growth, while gibberellins will counteract this germination inhibition. A null hypothesis is a statement of no difference. In other words, plant hormones will have no effect on plant growth.

In order to support or refute this hypothesis, the scientists developed a simulation or experiment. In this scenario, they exposed corn seeds to differing hormone treatments:

1. No treatment
2. Abscisic acid
3. Gibberellins and abscisic acid

They monitored the seeds for fourteen days and created a box plot using the data.

Let's look at how box plots are used to analyze data.

Box plots are broken into five main parts that create four primary regions. Each region contains a quarter of the data in the plot. The plot is initially broken into two regions using the following values: the minimum value, the maximum value, and the median or average of the set. Next, the minimum and median values are divided by the median of the lower half of the data or the first/lower quartile. Likewise, the median and the maximum values are divided by the median of the upper half of the data set or third/upper quartile. If the box is bigger, then there is a greater variance or spread in the data. Also, if the whiskers are very long, then the data possesses outliers.

When data is analyzed in a box plot or chart, it can then be used to make conclusions on the experiment. These conclusions will either support or refute a hypothesis. It is important to note that hypothesis cannot be proven support can only be added for them or against them. Even scientific theories such as gravity are not proven: they are just supported by a wealth of experiments and knowledge.

This brings us to the basis of this question. We need to make a conclusion based on the results of a simulation. Initially, we can see that the treatments had some effect due to the differences between the abscisic acid plot and the control plot; therefore, any choice that said there was not an effect can be disregarded. In the graph, we can clearly see that abscisic acid inhibits plant growth compared to the control; therefore, we can say, "abscisic acid inhibits plant growth." When we compare the control group with the treatment group that received both gibberellins and abscisic acid, we can see that they are comparable. This means that we can infer the following: "gibberellins can counteract abscisic acid inhibition." The correct choice is: "Yes, abscisic acid inhibits plant growth and gibberellins can counteract growth inhibition."

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable:

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope-intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative, then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them. The following two graphs possess a positive and moderately positive trendline, respectively:

Two other graphs possess a negative and slightly negative trendline, respectively:

Last, one graph has a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We start to solve this problem by picking out the graph with obvious positive or negative trends and excluding them. The following two graphs possess a positive and moderately positive trendline, respectively:

Two other graphs possess a negative and slightly negative trendline, respectively:

Last, one graph has a near horizontal line, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.