In probability, which description correctly defines P(A and B)?

In probability, which description correctly defines P(A and B)?

• A. The probability that either A or B occurs
• B. The probability that both A and B occur
• C. The probability that A occurs alone
• D. The probability that B occurs alone

Answer: (B)

P(A and B) is the probability that both A and B occur on the same trial—the intersection of the two events. It captures the chance that you see A and B together, not just one or the other. This differs from describing the likelihood of "either A or B" (that would be the union, P(A ∪ B)). It also isn’t about A occurring alone (A ∩ not B) or B occurring alone (B ∩ not A). You can compute it with P(A ∩ B) = P(A)P(B|A) or P(B)P(A|B). If A and B are independent, it simplifies to P(A)P(B). A helpful intuition: if you want both conditions to hold, you’re looking for the overlap between A’s outcomes and B’s outcomes. For example, drawing the ace of spades from a standard deck has probability 1/52, which is the probability that both “ace” and “spade” occur in the same draw. If A and B can’t happen together (mutually exclusive), that probability is zero.

P(A and B) is the probability that both A and B occur on the same trial—the intersection of the two events. It captures the chance that you see A and B together, not just one or the other. This differs from describing the likelihood of "either A or B" (that would be the union, P(A ∪ B)). It also isn’t about A occurring alone (A ∩ not B) or B occurring alone (B ∩ not A).

You can compute it with P(A ∩ B) = P(A)P(B|A) or P(B)P(A|B). If A and B are independent, it simplifies to P(A)P(B). A helpful intuition: if you want both conditions to hold, you’re looking for the overlap between A’s outcomes and B’s outcomes. For example, drawing the ace of spades from a standard deck has probability 1/52, which is the probability that both “ace” and “spade” occur in the same draw. If A and B can’t happen together (mutually exclusive), that probability is zero.