Solveeit Logo
Login

Question

Question: Let p be the probability that a man aged x year will die in a year time. The probability that out of...

Let p be the probability that a man aged x year will die in a year time. The probability that out of n men A1,A2,A3,....,An{A_1},{A_2},{A_3},....,{A_n} ​each aged x years, A1{A_1}​ will die & will be the first to die, is
A) 1−pnn\dfrac{{1 - {p^n}}}{n}
B) pn\dfrac{p}{n}
C) p(1−p)n−1n\dfrac{{p{{(1 - p)}^{n - 1}}}}{n}
D) 1−(1−p)nn\dfrac{{1 - {{(1 - p)}^n}}}{n}

Explanation

Solution

Hint : To solve this problem, we will assume that the probability that A1, A2, A3,...,An{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 A1,{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 A1,A2,A3,....,An{A_1},{A_2},{A_3},....,{A_n} ​each aged x years, we need to find the probability that when will A1{A_1}​ die and who will be the first one to die.
Now let us suppose that Ei{E_i} be the event that Ai{A_i} ​will die in a year where,   i=1,2,3,4,..,n.\;i = 1,2,3,4,..,n.
So, the probability that A1, A2, A3,...,An{A_1},{\text{ }}{A_2},{\text{ }}{A_3},...,{A_n} ​ dies in a year =P(Ei)=p = P\left( {{E_i}} \right) = p
And then the probability that none of A1, A2, A3,...,An{A_1},{\text{ }}{A_2},{\text{ }}{A_3},...,{A_n} ​ dies in a year =(1−p),(1−p),...(1−p) = \left( {1 - p} \right),\left( {1 - p} \right),...\left( {1 - p} \right)
=(1−p)n= {\left( {1 - p} \right)^n}
Now the probability that at least one of A1,A2,A3,...,An{A_1},{A_2},{A_3},...,{A_n} dies in a year =1−(1−p)n = 1 - {\left( {1 - p} \right)^n}
We need to find that the A1{A_1}​ is the first one to die, so the probability that among n men A1{A_1}​ is the first one to die is 1n.\dfrac{1}{n}.
So, probability that out of n men A1{A_1} will die & will be the first to die, is
⇒1n[1−(1−p)n].\Rightarrow \dfrac{1}{n}\left[ {1 - {{\left( {1 - p} \right)}^n}} \right].
Thus, option (D) 1−(1−p)nn\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 A1, A2, A3,...,An{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.