Let p be the probability that a man aged x years will die in a year time. The probability that out of ' n ' men A1, A2, A3, ldots . . ., An each aged ' x ' years, A1 will die will be the first to die is :

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 A1, A2, A3, ldots . . ., An each aged ' x ' years, A1 will die will be the first to die is : (A) (1-pn/n) (B) ( p / n ) (C) ( p (1- p ) n -1/ n ) (D) (1-(1-p)n/n)

Tardigrade Admin · Accepted Answer

Probability that no body die in one year

= (1 - p)^{n}

∴

Probability that atleast one dies

= 1 - (1 - P)^{n}

. Since all the

n

persons are equally likely to die.
Hence probability that

A_{1}

is the 1 st person to

die = \frac{1 - ( 1 - p ) ^{n}}{n}

Alternatively
Probability that exactly one dies

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

Since all the

n

persons are equally likely to die
hence probability that

A_{1} dies = \frac{^{n} C _{1} p ( 1 - p ) ^{n - 1}}{n}

.......(1)
Probability that exactly two dies and

A_{1}

is to die first

= \frac{^{n} C _{2} p ^{2} ( 1 - p ) ^{n - 2} \cdot ^{n - 1} C _{1}}{^{n} C _{2}} \cdot \frac{1}{2}

[\frac{^{n - 1} C _{1}}{^{n} C _{2}}

is the probability of choosing

A_{1}

and anyone out of

A_{2}

to

A_{n}

and out of these two, proabability that

A_{1}

dies first

= \frac{1}{2}]

= \frac{^{n} C _{2} p ^{2} ( 1 - p ) ^{n - 2}}{n}

.....(2)
II ly Probability exactly 3 dies and

A_{1}

is to die first

= \frac{^{n} C _{3} p ^{3} ( 1 - p ) ^{n - 3} \cdot ^{a - 1} C _{2}}{^{n} C _{3}} \cdot \frac{1}{3}

= \frac{^{n} C _{3} p ^{3} ( 1 - p ) ^{3}}{n}

.....(3)
Hence Total probability (using

1 - p = q

)

= \frac{^{n} C _{0} q ^{n} + ^{n} C _{1} p q ^{n - 1} + ^{n} C _{2} p ^{2} q ^{n - 2} + \dots\dots\dots\dots . + ^{n} C _{n} p ^{n} - ^{n} C _{0} q ^{n}}{n}

= \frac{( q + p ) ^{n} - ( 1 - p ) ^{n}}{n} = \frac{1 - ( 1 - p ) ^{n}}{n}

Q. Let $p$ be the probability that a man aged $x$ years will die in a year time. The probability that out of ' $n$ ' men $A_{1}, A_{2}, A_{3}, \dots ..., A_{n}$ each aged ' $x$ ' years, $A_{1}$ will die \& will be the first to die is :

$\frac{1 - p ^{n}}{n}$

$\frac{p}{n}$

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

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

Q. Let p be the probability that a man aged x years will die in a year time. The probability that out of ' n ' men A1​,A2​,A3​,…...,An​ each aged ' x ' years, A1​ will die \& will be the first to die is :

n1−pn​

np​

np(1−p)n−1​

n1−(1−p)n​

Q. Let $p$ be the probability that a man aged $x$ years will die in a year time. The probability that out of ' $n$ ' men $A_{1}, A_{2}, A_{3}, \dots ..., A_{n}$ each aged ' $x$ ' years, $A_{1}$ will die \& will be the first to die is :

$\frac{1 - p ^{n}}{n}$

$\frac{p}{n}$

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

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