Question

Let A and B be two events with P(A and Bc) = 1. Find P(B).

Recall that Bc denotes the complement of the event B. a. 0 b. 0.25
c. 0.5 d. 0.81 e. 1

Answer 1

The probability of event B, given P(A and Bc) = 1, is 0 because the events A and B never occur together, proven by the fact that P(A and B) must be 0 for the total probability involving A's outcomes to equal 1.

Step-by-step explanation:

The student is asking for the probability of an event B given that the probability of another event A occurring alongside the complement of B (Bc) is 1. In probability theory, the sum of the probabilities of an event and its complement is always equal to 1. Hence, P(A and Bc) + P(A and B) must be equal to P(A), as B and Bc encompass all possibilities for event B when event A occurs. But, since P(A and Bc) = 1, and probabilities cannot exceed 1, P(A and B) must be 0. This means that events A and B never occur together.

Given that A and B are independent, P(A and B) = P(A) × P(B), which should equal 0 in this case. Since P(A) is not 0 according to the provided reference, P(B) must be 0 for the multiplication to result in 0. Therefore, the probability of event B occurring is 0.