Question

Let P denote the chance that a person aged x years can die during a year. The chance that out of n men \[{A_1}\],\[{A_2}\], \[{A_3}\]. And every aged X, \[{A_1}\] can die during a year and can be the primary to die is
A) \[\dfrac{1}{n}\left[ {1 - {{(1 - p)}^n}} \right]\]
B) \[\left[ {1 - {{(1 - p)}^n}} \right]\]
C) \[\left[ {\dfrac{1}{{(n - 1)}}} \right]\left[ {1 - {{(1 - p)}^n}} \right]\]
D) None of these

Answer 1

Hint: In this question, we have given the chances P that a person aged x years can die during a year, so, to do this type of problem. First of all, find the probability that no man dies in a year. After that, find the probability that at least one man dies in a year to get the desired answer.

Complete step by step solution: 
According to the question, we have given that the probability that a man will die in a year, is P. There are n men such as \[{A_1}\], \[{A_2}\] \[{A_3}\]and \[{A_n}\].
 the probability that all the n man will die in a year will be,
\[ \Rightarrow P \times P \times P........n\]
Therefore, the probability that no one will die in a year will be,
\[ \Rightarrow (1 - P) \times (1 - P) \times (1 - P)........n\]
Therefore,
\[ \Rightarrow {(1 - P)^n}\]
Now, the probability that at least one man dies in a year will be,
\[ \Rightarrow 1 - {(1 - P)^n}\]
If the \[{A_1}\]is the first man who will die first, then the probability is written as,
\[ \Rightarrow \dfrac{1}{n}\]
 Therefore, the probability that out of n man \[{A_1}\]is the first man who will die first, is written as,
\[ \Rightarrow \dfrac{1}{n}\left[ {1 - {{(1 - P)}^n}} \right]\]
Now, the final answer is \[\dfrac{1}{n}\left[ {1 - {{(1 - P)}^n}} \right]\]
Therefore, the correct option is (A).

Note: The first point is to keep in mind that if the probability of an event is not given, then we assume the probability of the event. But in this question, the probability of an event is already given. So, there is no need to assume the probability of an event.