What is the difference between P(A or B) and P(A and B) in probability?

What is the difference between P(A or B) and P(A and B) in probability?

• A. P(A or B) = P(A) + P(B) - P(A and B); P(A and B) is the intersection
• B. P(A or B) equals P(A) × P(B); P(A and B) is the union
• C. P(A or B) equals P(A) + P(B); P(A and B) is the complement
• D. P(A or B) equals P(A) − P(B); P(A and B) is the sum

Answer: (A)

The key idea is how union and intersection relate in probability. P(A or B) is the probability that at least one of A or B occurs, while P(A and B) is the probability that both occur—the overlap between the two events. When you add P(A) and P(B), you count the overlap twice, so you subtract P(A and B) once to avoid double-counting. That gives P(A or B) = P(A) + P(B) − P(A and B). The term P(A and B) is the intersection of A and B. For a concrete example, think of a single card draw. Let A be “ace” and B be “heart.” Then P(A) = 4/52, P(B) = 13/52, and P(A and B) = 1/52 (the ace of hearts). So P(A or B) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13. If A and B were disjoint, the overlap would be zero and P(A or B) would simply be P(A) + P(B).

The key idea is how union and intersection relate in probability. P(A or B) is the probability that at least one of A or B occurs, while P(A and B) is the probability that both occur—the overlap between the two events. When you add P(A) and P(B), you count the overlap twice, so you subtract P(A and B) once to avoid double-counting. That gives P(A or B) = P(A) + P(B) − P(A and B). The term P(A and B) is the intersection of A and B.

For a concrete example, think of a single card draw. Let A be “ace” and B be “heart.” Then P(A) = 4/52, P(B) = 13/52, and P(A and B) = 1/52 (the ace of hearts). So P(A or B) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.

If A and B were disjoint, the overlap would be zero and P(A or B) would simply be P(A) + P(B).