Question

Find the probability that exactly two of them are selected.

Hint:

For "exactly two" events, consider all pairs of selections and one failure, and sum their probabilities.

Answer 1

Step 1: Compute the probability of exactly two being selected
The probability of exactly two being selected is: \[ P(\text{Exactly two selected}) = P(R) \cdot P(J) \cdot P(\overline{A}) + P(R) \cdot P(\overline{J}) \cdot P(A) + P(\overline{R}) \cdot P(J) \cdot P(A). \] Substitute the values: \[ P(\text{Exactly two selected}) = \frac{1}{5} \cdot \frac{1}{3} \cdot \frac{3}{4} + \frac{1}{5} \cdot \frac{2}{3} \cdot \frac{1}{4} + \frac{4}{5} \cdot \frac{1}{3} \cdot \frac{1}{4}. \] Simplify each term: \[ P(\text{Exactly two selected}) = \frac{3}{60} + \frac{2}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20}. \] Final Result: The probability that exactly two of them are selected is \( \frac{3}{20} \).