A man and a woman appear in an interview for two vacancies in the same post. The probability of mans selection of 1/4 and that the womans selection is 1/3. What is the probability that none of them will be selected?

Explanation

To solve the problem, we need to find the probability that neither the man nor the woman will be selected for the vacancies.

### Step-by-Step Solution:

1. **Identify the probabilities of selection:**
- The probability of the man being selected (P(E1)) is given as \( \frac{1}{4} \).
- The probability of the woman being selected (P(E2)) is given as \( \frac{1}{3} \).

2. **Calculate the probabilities of not being selected:**
- The probability that the man is not selected (P(E1 complement)) is:
\[ P(E1') = 1 - P(E1) = 1 - \frac{1}{4} = \frac{3}{4} \]
- The probability that the woman is not selected (P(E2 complement)) is:
\[ P(E2') = 1 - P(E2) = 1 - \frac{1}{3} = \frac{2}{3} \]

3. **Determine the probability that neither is selected:**
- Since the events of selection are independent, the probability that neither the man nor the woman is selected (P(E1' ∩ E2')) can be calculated by multiplying the probabilities of their complements:
\[ P(E1' ∩ E2') = P(E1') \times P(E2') = \frac{3}{4} \times \frac{2}{3} \]

4. **Perform the multiplication:**
- Calculate:
\[ P(E1' ∩ E2') = \frac{3 \times 2}{4 \times 3} = \frac{6}{12} = \frac{1}{2} \]

5. **Conclusion:**
- The probability that none of them will be selected is \( \frac{1}{2} \).

### Final Answer:
The probability that neither the man nor the woman will be selected is \( \frac{1}{2} \).

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