Question

Let $P$ denotes 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

• A. $\frac{1}{n[1-{(1-p)}^n]}$
• B. $[1-{(1-p)}^n]$
• C. $\frac{1}{(n-1)}[1-{(1-p)}^n]$
• D. None of these

Answer 1

The correct option is A

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

Explanation for the correct option:

Finding the required probability.

Given: $P$ denotes the probability that a man aged $x$ years will die in a year.

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 $A1,A2,A3,...,An$ 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.

$$\begin{gathered} =(1-p).(1-p),...(1-p) \\ ={(1-p)}^n \end{gathered}$$

Now the probability that at least one of those dies is $A_1,A_2,A_3,...,A_n=1-{(1-p)}^n$​

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

Therefore, the required probability is

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

Hence the correct option is (A).