Events \( A \) and \( B \) are mutually exclusive. Suppose event \( A \) occurs with probability 0.48 and event \( B \) occurs with probability 0.23 . Compute the following. (If necessary, consult a list of formulas.) (a) Compute the probability that \( A \) occurs but \( B \) does not occur. (b) Compute the probability that either \( A \) occurs without \( B \) occurring or \( B \) occurs without \( A \) occurring. \( \square \)

Let's analyze the given information and solve each part step by step. **Given:** - Events \( A \) and \( B \) are **mutually exclusive**, which means they cannot occur simultaneously. Formally, \( P(A \cap B) = 0 \). - \( P(A) = 0.48 \) - \( P(B) = 0.23 \) ### (a) Probability that \( A \) occurs but \( B \) does not occur. Since \( A \) and \( B \) are mutually exclusive, if \( A \) occurs, \( B \) **cannot** occur. Therefore, the probability that \( A \) occurs **without** \( B \) occurring is simply the probability of \( A \): \[ P(A \text{ and not } B) = P(A) = 0.48 \] **Answer for (a):** \( 0.48 \) ### (b) Probability that either \( A \) occurs without \( B \) occurring or \( B \) occurs without \( A \) occurring. This probability can be expressed as: \[ P\big[(A \text{ and not } B) \text{ or } (B \text{ and not } A)\big] \] Since \( A \) and \( B \) are mutually exclusive: - \( P(A \text{ and not } B) = P(A) = 0.48 \) - \( P(B \text{ and not } A) = P(B) = 0.23 \) Because these two events (\( A \) occurring without \( B \), and \( B \) occurring without \( A \)) are also mutually exclusive, their probabilities can be added directly: \[ P\big[(A \text{ and not } B) \text{ or } (B \text{ and not } A)\big] = P(A) + P(B) = 0.48 + 0.23 = 0.71 \] **Answer for (b):** \( 0.71 \) ### Summary: - **(a)** The probability that \( A \) occurs but \( B \) does not is **0.48**. - **(b)** The probability that either \( A \) occurs without \( B \) or \( B \) occurs without \( A \) is **0.71**. **Final Answers:** - **(a)** \( 0.48 \) - **(b)** \( 0.71 \)