How to find probability distributions for discrete random variables - AP Statistics
Card 0 of 6
During a week's worth of soccer practice, a player practices 
total free kicks and has a 
chance of scoring. What is the probability that he or she scored at least 
times? Assume each shot is independent.
During a week's worth of soccer practice, a player practices
Two steps are crucial here.
First, we need to recognize this is a binomial distribution with 
and 
.
Second, we need to realize we can use a normal approximation of the binomial since 
and 
, which are both larger than 5.
With that said, we can calculate a 
-score and its 
-value, keeping in mind that our mean will be 
and our standard deviation will be 
, which is about 
.
Two steps are crucial here.
First, we need to recognize this is a binomial distribution with
Second, we need to realize we can use a normal approximation of the binomial since
With that said, we can calculate a
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Which of the following would be considered a binomial experiment?
Which of the following would be considered a binomial experiment?
There are four conditions that need to be satisfied for a binomial experiment:
-
Each trial must have two outcomes.
-
Each trial must be independent.
-
All trials must be identical.
-
The probabilities of the outcomes remain constant must not change with each trial.
The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.
There are four conditions that need to be satisfied for a binomial experiment:
-
Each trial must have two outcomes.
-
Each trial must be independent.
-
All trials must be identical.
-
The probabilities of the outcomes remain constant must not change with each trial.
The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.
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Suppose you are throwing three darts and you have a one third chance of hitting the bull's eye. Each throw is independent of one another. What is the chance of hitting the bull's eye at least once?
Suppose you are throwing three darts and you have a one third chance of hitting the bull's eye. Each throw is independent of one another. What is the chance of hitting the bull's eye at least once?
To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).
So we have P(at least one bull's eye)=1-P(no bull's eye).
The chance of getting no bull's eyes is 
.
This means the probability of getting at least one bull's eye is
To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).
So we have P(at least one bull's eye)=1-P(no bull's eye).
The chance of getting no bull's eyes is
This means the probability of getting at least one bull's eye is
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A particle travels left with probability one sixth and right with probability five sixths. Each movement is independent of the others. What is the chance that after three movements, the particle ends up one unit to the right?
A particle travels left with probability one sixth and right with probability five sixths. Each movement is independent of the others. What is the chance that after three movements, the particle ends up one unit to the right?
The movements that this particle can make include: RRL, RLR, LRR.
The chance of getting RRL is 
. This is also the chance of getting any of those movements.
To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.
Thus, we get the following as the solution.
The movements that this particle can make include: RRL, RLR, LRR.
The chance of getting RRL is
To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.
Thus, we get the following as the solution.
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If you flip a biased coin, which has a 
chance of being heads and 
of being tails, until you get a head, what is the chance that it takes five flips until you get a head?
If you flip a biased coin, which has a
To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.
Each tail has a prob. of 
and a head is 
, so we multiply 
to the power of 4 (because we need 4 tails) by 
(for the single head).
So the probability is
.
To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.
Each tail has a prob. of
So the probability is
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Which of the following is a discrete random variable?
Which of the following is a discrete random variable?
A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.
A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.
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