Question

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

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

Answer 1

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

Let $E_i$ be the event that $A_i$ will die in a year where i=1, 2, 3, 4, ..., n

The probability that none of $A_1,A_2,A_3,...,A_n$ dies in a year

=$(1-p_{A_1}).(1-p_{A_2}),...,(1-p_{A_n})$

Where $p_{A_1},p_{A_2},...,p_{A_n}$ are the probabilities that $A_1,A_2,A_3,...,A_n$ dies in a year respectively.

=$(1-p).(1-p),...(1-p)=(1-p{)}^n$

Now the probability that at least one of $A_1,A_2,A_3,...,A_n$

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

The probability that among n men $A_1$ is the first one to die is $\frac{1}{n}$

Therefore, the required probability is

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