The probability that a man will be alive for 10 more years is `1/4` and the probability that his wife will alive for 10 more years is `1/3`. The probability that none of them will be alive for 10 more years, is

Options

A. `5/12`
B. `1/2`
C. `7/12`
D. `11/12`

Answer

To solve the problem, we need to find the probability that neither the man nor his wife will be alive in 10 years. We can do this by first calculating the probabilities that each of them will not be alive and then multiplying these probabilities together.

### Step-by-Step Solution:

1. **Identify the Given Probabilities:**
- Probability that the man will be alive for 10 more years, \( P(M) = \frac{1}{4} \).
- Probability that the wife will be alive for 10 more years, \( P(W) = \frac{1}{3} \).

2. **Calculate the Probability that the Man Will Not Be Alive:**
- The probability that the man will not be alive for 10 more years is given by:
\[ P(\text{not } M) = 1 - P(M) = 1 - \frac{1}{4} = \frac{3}{4}. \]

3. **Calculate the Probability that the Wife Will Not Be Alive:**
- The probability that the wife will not be alive for 10 more years is given by:
\[ P(\text{not } W) = 1 - P(W) = 1 - \frac{1}{3} = \frac{2}{3}. \]

4. **Calculate the Probability that Neither Will Be Alive:**
- Since the events are independent, the probability that neither the man nor the wife will be alive is:
\[ P(\text{not } M \text{ and not } W) = P(\text{not } M) \times P(\text{not } W) = \frac{3}{4} \times \frac{2}{3}. \]

5. **Perform the Multiplication:**
- Multiply the fractions:
\[ P(\text{not } M \text{ and not } W) = \frac{3 \times 2}{4 \times 3} = \frac{6}{12} = \frac{1}{2}. \]

6. **Conclusion:**
- Therefore, the probability that neither the man nor his wife will be alive for 10 more years is \( \frac{1}{2} \).

### Final Answer:
The probability that none of them will be alive for 10 more years is \( \frac{1}{2} \).