Question

Given P(A) = 0.8 and P(B) = 0.4,do the following.

(a) If A and B are independent events, compute P(A and B).
(b) If P(A | B) = 0.1, compute P(A and B).

Answer 1

For independent events A and B, P(A and B) is the product of their individual probabilities. If A and B are independent, then P(A and B) = 0.32. If P(A | B) = 0.1, then P(A and B) = 0.04.

Step-by-step explanation:

Probability of Independent Events

To answer your questions about the probability of events A and B:

• (a) If A and B are independent events, the probability of A and B occurring together, denoted by P(A and B), is the product of their individual probabilities. Therefore, P(A and B) = P(A) × P(B) = 0.8 × 0.4 = 0.32.

• (b) If P(A | B) = 0.1, this represents the probability of A given that B has occurred. To compute P(A and B), we use the definition of conditional probability: P(A and B) = P(A | B) × P(B) = 0.1 × 0.4 = 0.04.

To determine if events A and B are independent, we would check if P(A and B) = P(A)P(B). In case (a), the definition is satisfied, so A and B are independent. In case (b), since P(A | B) != P(A), it suggests that A and B are not independent.