Question

Let p be the probability that a man aged x year 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) $\dfrac{{1 - {p^n}}}{n}$
B) $\dfrac{p}{n}$
C) $\dfrac{{p{{(1 - p)}^{n - 1}}}}{n}$
D) $\dfrac{{1 - {{(1 - p)}^n}}}{n}$

Answer 1

Hint: To solve this problem, we will assume that the probability that \[{A_1},{\text{ }}{A_2},{\text{ }}{A_3},...,{A_n}\] ​ dies in a year is p, afterwards we will find the probability of all n men dies in a year, then we will find the probability of none dies in a year, now with that we will find the probability that at least one man dies and that too \[{A_1},\] and hence with that we will get our required answer.

Complete step-by-step answer:
We have been given that p is the probability that a man aged x year will die in a year's time. It is given the probability that out of n men \[{A_1},{A_2},{A_3},....,{A_n}\] ​each aged x years, we need to find the probability that when will \[{A_1}\]​ die and who will be the first one to die.
Now let us suppose that \[{E_i}\] be the event that \[{A_i}\] ​will die in a year where, \[\;i = 1,2,3,4,..,n.\]
So, the probability that \[{A_1},{\text{ }}{A_2},{\text{ }}{A_3},...,{A_n}\] ​ dies in a year \[ = P\left( {{E_i}} \right) = p\]
And then the probability that none of \[{A_1},{\text{ }}{A_2},{\text{ }}{A_3},...,{A_n}\] ​ dies in a year \[ = \left( {1 - p} \right),\left( {1 - p} \right),...\left( {1 - p} \right)\]
\[ = {\left( {1 - p} \right)^n}\]
Now the probability that at least one of \[{A_1},{A_2},{A_3},...,{A_n}\] dies in a year \[ = 1 - {\left( {1 - p} \right)^n}\]
We need to find that the \[{A_1}\]​ is the first one to die, so the probability that among n men \[{A_1}\]​ is the first one to die is $\dfrac{1}{n}.$
So, probability that out of n men \[{A_1}\] will die & will be the first to die, is
 \[\Rightarrow \dfrac{1}{n}\left[ {1 - {{\left( {1 - p} \right)}^n}} \right].\]
Thus, option (D) $\dfrac{{1 - {{(1 - p)}^n}}}{n}$ is correct.
So, the correct answer is “Option D”.

Note: Students should note that in these types of questions, you will need to assume a few values. Just like we have assumed that the probability that \[{A_1},{\text{ }}{A_2},{\text{ }}{A_3},...,{A_n}\] ​ dies in a year p, it is done because we saw the options, and then that the assumed value should be p.