Conditional Probability
Big Idea: Conditional probability answers the question: "How does knowing that one event has occurred change the probability of another event?"
1. Formal Definition
The conditional probability of event $A$ given that event $B$ has occurred is defined as:
$$ P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0 $$
We read $P(A \mid B)$ as "the probability of $A$ given $B$."
- $P(A \cap B)$ is the probability that both $A$ and $B$ occur.
- $P(B)$ is the probability that event $B$ occurs (our new "reduced" sample space).