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