Which expression correctly defines P(A|B)?

Which expression correctly defines P(A|B)?

• A. P(B|A)
• B. P(A∩B)/P(B)
• C. P(A) - P(B)
• D. P(A)∪B

Answer: (B)

Conditional probability asks: what is the chance that A happens when we already know B has occurred? It is defined as the probability that both A and B occur, divided by the probability that B occurs, provided B has positive probability. So the expression that matches is the ratio P(A∩B) / P(B), with the condition P(B) > 0. Think of narrowing the sample space to the outcomes where B happens, and then asking what fraction of those outcomes also have A. If you tried to use P(B|A), you’d be measuring the chance of B given A, which is a different condition. Subtracting probabilities or taking the union of A and B doesn’t reflect conditioning either.

Conditional probability asks: what is the chance that A happens when we already know B has occurred? It is defined as the probability that both A and B occur, divided by the probability that B occurs, provided B has positive probability. So the expression that matches is the ratio P(A∩B) / P(B), with the condition P(B) > 0.

Think of narrowing the sample space to the outcomes where B happens, and then asking what fraction of those outcomes also have A. If you tried to use P(B|A), you’d be measuring the chance of B given A, which is a different condition. Subtracting probabilities or taking the union of A and B doesn’t reflect conditioning either.