Conditional Probability

Conditional probability is the probability of A given that B has already occurred: [formula] In a two-way table, this becomes: [formula] The key word is "given" — it restricts your sample space.

Example: In a class: 60% study math, 40% study science, 25% study both. Given that a student studies math, what is the probability they also study science?

Two events are independent if knowing one doesn't change the probability of the other: [formula] Equivalently: P(A text{ and } B) = P(A) times P(B) If these equalities don't hold, the events are dependent (associated).

Example: Event A: P(A) = 0.5. Event B: P(B) = 0.3. P(A text{ and } B) = 0.15. Are A and B independent?