Let A, B, C be 3 independent events such that `P(A)=1/3,P(B)=1/2,P(C)=1/4`. Then find the probability of exactly 2 events occurring out of 3 events.

Options

A. option 1 = 1/4
B. option 2 = 9/24
C. option 3 = 3/4
D. option 4 = None of these

Answer

To find the probability of exactly 2 out of the 3 independent events A, B, and C occurring, we can follow these steps:

### Step 1: Identify the probabilities of each event and their complements
Given:
- \( P(A) = \frac{1}{3} \)
- \( P(B) = \frac{1}{2} \)
- \( P(C) = \frac{1}{4} \)

The probabilities of the complements (not occurring) are:
- \( P(A') = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \)
- \( P(B') = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2} \)
- \( P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4} \)