The probability that a man aged 50 years will die in a year is p. The probability that out of n men `A_(1)`, `A_(2)`, `A_(3)`, ….., `A_(n)` each aged 50 year, `A_(1)` will die and first to die is

Options

A. `(1-(1-p)^(n))/n`
B. `(1-(1-p)^(n))/(n^(2))`
C. `1-(1-p)^(n)`
D. `[1-(1-p)^(n)]^(2)/(n^(2))`

Answer

To solve the problem, we need to find the probability that out of \( n \) men, each aged 50 years, the man \( A_1 \) will be the first to die within a year, given that the probability of a man aged 50 dying in a year is \( p \).

### Step-by-Step Solution:

1. **Understanding the Probability of Death**:
- The probability that a man aged 50 will die in a year is given as \( p \).
- Therefore, the probability that a man aged 50 will survive the year is \( 1 - p \).

2. **Finding the Probability of Survival for All Men**:
- If we have \( n \) men, the probability that all \( n \) men survive the year is given by:
\[ (1 - p)^n \]

3. **Finding the Probability that At Least One Man Dies**:
- The probability that at least one man dies out of \( n \) men is the complement of the probability that all survive:
\[ P(\text{at least one dies}) = 1 - (1 - p)^n \]

4. **Finding the Probability that \( A_1 \) is the First to Die**:
- For \( A_1 \) to be the first to die, \( A_1 \) must die, and all other \( n-1 \) men must survive.
- The probability that \( A_1 \) dies is \( p \).
- The probability that each of the other \( n-1 \) men survives is \( (1 - p)^{n-1} \).
- Therefore, the probability that \( A_1 \) is the first to die is:
\[ P(A_1 \text{ is first to die}) = p \cdot (1 - p)^{n-1} \]

5. **Combining the Probabilities**:
- The total probability that \( A_1 \) will die and be the first to die among \( n \) men is:
\[ P(A_1 \text{ dies first}) = p \cdot (1 - p)^{n-1} \]

### Final Answer:
Thus, the required probability that \( A_1 \) will die and be the first to die is:
\[ P(A_1 \text{ is the first to die}) = p \cdot (1 - p)^{n-1} \]