Question

Odds are $8$ to $5$ against a person who is $40$yr old living till he is $70$ and $4$ to $3$ against another person now $50$ till he will be living $80$. Probability that one of them will be alive next $30$yr. is-
(A) $\dfrac{{59}}{{91}}$
(B) $\dfrac{{44}}{{91}}$
(C) $\dfrac{{51}}{{91}}$
(D) $\dfrac{{32}}{{91}}$

Answer 1

First we have to find the probability for the 2 persons to be alive and dead for the next 30 years. Later we must find the probability that one of them will be alive for the next 30 years.
In any random test or experiment, the sum of probabilities of happening and not happening an event is always equal to $1$, i.e., $P\left( E \right) + P\left( {\overline E } \right) = 1$.

Complete step-by-step answer:
Let $A$ be the event that person A will live in the next thirty years and let $B$ be the event that person B will live in the next thirty years.
We have given, odds are $8$ to $5$ against person A that he will live next thirty years.
So, the probability that the person A will be alive in next $30$ years, $P\left( A \right)$ $= \dfrac{5}{{5 + 8}}$ $= \dfrac{5}{{13}}$
The probability that the person A will be dead in next $30$ years, $P\left( {\overline A } \right) = 1 - P\left( A \right) = 1 - \dfrac{5}{{13}} = \dfrac{8}{{13}}$
Now for second person B, odds are $8$ to $5$ against that he will live next thirty years.
So, the probability that the person B will be alive in next $30$ years, $\;P\left( B \right) = \dfrac{3}{{3 + 4}} = \dfrac{3}{7}$
The probability that the person A will be dead in next $30$ years, $P\left( {\overline B } \right) = 1 - P\left( B \right) = 1 - \dfrac{3}{7} = \dfrac{4}{7}$
There are two ways in which one person is alive after $30$ years, i.e., $A\overline B$ and $\overline A B$.
So, required probability$= P\left( A \right) \cdot P\left( {\overline B } \right) + P\left( {\overline A } \right) \cdot P\left( B \right)$
$= \dfrac{5}{{13}} \times \dfrac{4}{7} + \dfrac{8}{{13}} \times \dfrac{3}{7}$
$= \dfrac{{20}}{{91}} + \dfrac{{24}}{{91}}$
$= \dfrac{{44}}{{91}}$

The required probability is $= \dfrac{{44}}{{91}}$.

Note: An independent event is one where the result does not get impacted by some other events. Here, $A\overline B$ and $\overline A B$ are two independent events, so the required probability is given by $P\left( A \right) \cdot P\left( {\overline B } \right) + P\left( {\overline A } \right) \cdot P\left( B \right)$.