Which statement about the joint probability P(A ∩ B) is true?

Which statement about the joint probability P(A ∩ B) is true?

• A. The probability that B occurs given A.
• B. The probability that A does not occur.
• C. The probability that either A or B occurs.
• D. The probability that both A and B occur.

Answer: (D)

The key idea is what the intersection A ∩ B represents in probability. P(A ∩ B) is the probability that both events happen together. It describes the chance that A and B occur at the same time. This is different from the probability of B given A, which is P(B|A) and equals P(A ∩ B) divided by P(A) (when P(A) > 0). It’s also different from the probability that A does not occur, which is P(A^c), and from the probability that either A or B occurs, which is the union P(A ∪ B). So, P(A ∩ B) specifically captures the event that both A and B occur, making it the joint or intersection probability. For example, if P(A) = 0.6 and P(A ∩ B) = 0.3, then the chance that both A and B occur is 0.3, and the chance that B occurs given A is 0.3/0.6 = 0.5.

The key idea is what the intersection A ∩ B represents in probability. P(A ∩ B) is the probability that both events happen together. It describes the chance that A and B occur at the same time.

This is different from the probability of B given A, which is P(B|A) and equals P(A ∩ B) divided by P(A) (when P(A) > 0). It’s also different from the probability that A does not occur, which is P(A^c), and from the probability that either A or B occurs, which is the union P(A ∪ B).

So, P(A ∩ B) specifically captures the event that both A and B occur, making it the joint or intersection probability. For example, if P(A) = 0.6 and P(A ∩ B) = 0.3, then the chance that both A and B occur is 0.3, and the chance that B occurs given A is 0.3/0.6 = 0.5.