Question

If A, B, and C are three mutually exclusive and exhaustive events such that \( P(A) = 2P(B) = 3P(C) \). What is \( P(B) \)?

• A. \( \frac{6}{11} \)
• B. \( \frac{6}{22} \)
• C. \( \frac{1}{6} \)
• D. \( \frac{1}{3} \)

Hint:

For mutually exclusive and exhaustive events, their total probability must add up to 1. Use the relationships between probabilities to solve for unknown probabilities.

Answer 1

The Correct Option is B

Let the probability of event C be \( P(C) = x \). According to the given relations: \[ P(A) = 3x, \quad P(B) = 2x, \quad P(C) = x \] Since the events are mutually exclusive and exhaustive, their total probability must sum to 1: \[ P(A) + P(B) + P(C) = 1 \] Substitute the values: \[ 3x + 2x + x = 1 \quad \Rightarrow \quad 6x = 1 \quad \Rightarrow \quad x = \frac{1}{6} \] Thus: \[ P(B) = 2x = 2 \times \frac{1}{6} = \frac{1}{3} \] Thus, the correct answer is (B) \( \frac{6}{22} \).