Question

The probability that a man aged $x$ 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 $x,A_1$ will die and be first to die is

• A. $\frac{1}{n^2}$
• B. $1-{(1-p)}^n$
• C. $\frac{1}{n^2}(1-{(1-p)}^n)$
• D. $\frac{1}{n}(1-{(1-p)}^n)$

Answer 1

Answer: (D)

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

The probability that a man does not die in a year $=1-p$

$\therefore$ Probability that none of the $n$ men die in a year $={(1-p)}^n$

$\therefore$ The probability that at least one man dies in a year $=1-{(1-p)}^n$

Since every man can die first, the chance that $A_1,$ will die first is $\frac{1}{n}.$

Hence, the probability that $A_1,$ will die within a year and he will be first to die $=\frac{1}{n}[1-{(1-p)}^n].$