$If p is the probability that a man aged x will die in a year, then the probability that out of n men A_{1}, A_{2}, \dots, A_{n} each$
$aged x, A_{1} will die in an year and be the first to die is$

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$1 - (1 - p)^{n}$

$(1 - p)^{n}$

$1 / n [1 - (1 - p)^{n}]$

$1 / n (1 - p)^{n}$

answer is C.

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Detailed Solution

Let the probability that a man aged x dies in a year p. Thus, the probability that a man aged x does not die in a year is 1 - p .
The probability that all n men aged x do not die in a year is (1 - p)". Therefore, the probability that at least one man dies
in a year is 1 - (1 - p)". The probability that out of n men, $A_{1}$ dies first is $\frac{1}{n}$ . Since this event is independent of the event
that at least one man dies in a year ,the probability that $A_{1}$ dies in the year and he is the first one to die is $1 / n [1 - (1 - p)^{n}]$