Probability Theory explores the mathematical framework for quantifying uncertainty and making informed decisions based on random events.
Conditional probability answers questions like: "What's the chance of event \(A\) happening, given that event \(B\) already happened?" It's written as \(P(A|B)\).
The formula is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
where \(P(A \cap B)\) is the probability that both \(A\) and \(B\) occur.
Conditional probability helps us make better predictions by using extra information. It's used in everything from medical testing to online recommendations!
\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]
If 30% of students play soccer and 10% play both soccer and basketball, the probability a soccer player also plays basketball is 10/30 = 1/3.
The probability of it raining, given that it's cloudy, is higher than the probability of rain in general.
Conditional probability measures the chance of something happening, knowing that something else has happened.