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(A) Let $E_i$ denote the event that $A_i$ dies in a year. Then $P(E_i) = p$ and $P(E'_i) = 1 - p$ for $i = 1, 2, ..., n$. The probability that none of

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

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(A) Let $E_i$ denote the event that $A_i$ dies in a year. Then $P(E_i) = p$ and $P(E'_i) = 1 - p$ for $i = 1, 2, ..., n$. The probability that none of $A_1, A_2, ..., A_n$ dies in a year is $P(E'_1 \cap E'_2 \cap ... \cap E'_n) = P(E'_1)P(E'_2)...P(E'_n) = (1 - p)^n$,since $E_1, E_2, ..., E_n$ are independent. Let $E$ denote the event that at least one of $A_1, A_2, ..., A_n$ dies in a year. Then $P(E) = 1 - P(E'_1 \cap E'_2 \cap ... \cap E'_n) = 1 - (1 - p)^n$. Let $F$ denote the event that $A_1$ is the first to die. Since all men have the same age $x$,the probability that any specific person is the first to die among those who die is $\frac{1}{n}$. Thus,$P(F|E) = \frac{1}{n}$. Therefore,the probability that $A_1$ dies and is the first to die is $P(F) = P(E) 	imes P(F|E) = \frac{1}{n} [1 - (1 - p)^n]$.

Let $p$ denote the probability that a man aged $x$ years will die in a year. The probability that out of $n$ men $A_1, A_2, A_3, ..., A_n$ each aged $x$,$A_1$ will die in a year and will be the first to die,is

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