Problem 8 - A First Course in Probability

Question

Suppose that \(A\) and \(B\) are mutually exclusive events for which \(P(A)=.3\) and \(P(B)=.5 .\) What is the probability that (a) either \(A\) or \(B\) occurs? (b) \(A\) occurs but \(B\) does not? (c) both \(A\) and \(B\) occur?

Step-by-Step Solution

Answer

(a) The probability that either \(A\) or \(B\) occurs is \(0.8\). (b) The probability that \(A\) occurs but \(B\) does not is \(0.3\). (c) The probability that both \(A\) and \(B\) occur is \(0\).

Step 1: (a) Probability of A or B occurring

To find the probability of either \(A\) or \(B\) occurring, we will use the formula for the probability of the union of two events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Since \(A\) and \(B\) are mutually exclusive, \(P(A \cap B) = 0\). Therefore, the formula simplifies to \(P(A \cup B) = P(A) + P(B)\). Using the given probabilities, we obtain: \(P(A \cup B) = P(A) + P(B) = 0.3 + 0.5 = 0.8\) So the probability that either \(A\) or \(B\) occurs is \(0.8\).

Step 2: (b) Probability of A occurring but not B

Since A and B are mutually exclusive events, the probability of A occurring but not B is just the probability of A occurring. Therefore: \(P(A \text{ and not } B) = P(A) = 0.3\) So the probability that \(A\) occurs but \(B\) does not is \(0.3\).

Step 3: (c) Probability of both A and B occurring

As mentioned earlier, since \(A\) and \(B\) are mutually exclusive events, they cannot both happen at the same time. Therefore, the probability that both \(A\) and \(B\) occur is 0: \(P(A \cap B) = 0\) So the probability that both \(A\) and \(B\) occur is \(0\).