Question

The probability that a man aged $x$ years will die in a year is ${}^{\prime }{p}^{\prime }.$ The probability that out of ${}^{\prime }{n}^{\prime }$ men $M_1,M_2,...M_n,$ each aged years $x,M_1$ will die and be the 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

The correct option is D $\frac{1}{n}[1-{(1-p)}^n]$

Probability that a man aged $x$ dies in a year $=p.$
Therefore the probability that at least one man dies in a year $=1-{(1-p)}^{n}i.e.,1-p$ (no man dies in a year).
Probability that out of $n$ men, $M_1$ is first to die $=\frac{1}{n}$ since this event is independent.
Hence, the probability that $M_1$ dies in the year and he is first to die $=\frac{1-{(1-p)}^n}{n}$