Which of the following is the joint probability?

• A. P(A) + P(B)
• B. P(A ∩ B)
• C. P(A|B)
• D. P(B|A)

Answer: (B)

Joint probability is the probability that both events occur at the same time. It’s captured by the intersection of the two events, written as P(A ∩ B). That quantity directly measures the chance that A and B happen together.

The other expressions mix probabilities in different ways. P(A) + P(B) adds the chances of A and B separately and isn’t the probability of both happening (it reflects the sum of their individual chances and, if A and B can occur together, double-counts that overlap). Conditional probabilities, like P(A|B), describe the chance of A given that B has occurred and relate to the joint via P(A ∩ B) = P(A|B)P(B). Similarly, P(B|A) relates through P(A ∩ B) = P(B|A)P(A). So the one that directly represents the probability of both events happening is P(A ∩ B). For example, if P(A) = 0.5, P(B) = 0.4, and their overlap P(A ∩ B) = 0.2, the joint probability is 0.2, while the other expressions would give different values.