Module 2 · Probability Foundations · Topic 7
Independent Events. Two events are independent if the occurrence of one does not change the probability of the other. Formally, A and B are independent if P(A | B) = P(A).
A and B are independent if and only if P(A ∩ B) = P(A) × P(B)
P(A | B) = P(A and B) / P(B) — Probability of A after restricting attention to cases where B occurred.P(A | B) = P(B | A)P(A) / P(B) — Updates a prior probability after observing evidence.Roll a die and flip a coin. P(6 and heads) = P(6) × P(heads) = (1/6)(1/2) = 1/12. The die roll does not affect the coin flip, so they are independent.
Don’t confuse these! Mutually exclusive events cannot both happen. When both events have positive probability, they are dependent. Independent events can both happen - one just doesn’t influence the other.
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