Conditional probability

Japanese school year: Math A

What you learn

This topic covers calculating conditional probabilities, which determine the likelihood of an event occurring given that another event has already happened. It is widely used in medical testing and predictive models to update probabilities with new evidence. A solid understanding of basic probability and set operations is recommended.

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Key points

This formula calculates the conditional probability $P_A(B)$ of event $B$ occurring given that event $A$ has occurred. Divide the joint probability $P(A \cap B)$ by the condition's probability $P(A)$ .

P_A(B) = \frac{P(A \cap B)}{P(A)}

This multiplication rule finds the probability that both events $A$ and $B$ happen. Multiply the probability of $A$ by the conditional probability $P_A(B)$ of $B$ given $A$ .

P(A \cap B) = P(A)P_A(B)

This formula finds conditional probability directly from the number of outcomes. Treat the outcome count $n(A)$ of the condition as the new total, and divide favorable outcomes $n(A \cap B)$ by it.

P_A(B) = \frac{n(A \cap B)}{n(A)}

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Practice 1Coming soon