Question

The probability that a man will live 10 more years is $\dfrac{1}{4}$ and the probability that his wife will live 10 years is $\dfrac{1}{3}$. Then the probability that neither will be alive in 10 more years is

Answer 1

Hint:Find the probability that the man will not be alive in 10 more years using the information that the probability that the man will live 10 more years is $\dfrac{1}{4}$. Similarly, find the probability that his wife will not be alive in 10 more years. Multiply these probabilities to get the answer.

Formula Used: \[{\text{Probability of complement of an event = 1 - Probability of that event}}\],i.e., $P(\overline A ) = 1 - P(A)$

Complete step by step Solution:
Let the probability that the man will live 10 more years be $P(M) = \dfrac{1}{4}$
So, the probability that the man will not be alive in 10 more years is $P(\overline M ) = 1 - \dfrac{1}{4} = \dfrac{3}{4}$
Let the probability that his wife will live 10 more years be $P(W) = \dfrac{1}{3}$
So, the probability that his wife will not be alive in 10 more years is $P(\overline W ) = 1 - \dfrac{1}{3} = \dfrac{2}{3}$
The probability that neither, the man nor his wife, will be alive in 10 more years is $P(\overline M ).P(\overline W ) = \dfrac{3}{4}.\dfrac{2}{3} = \dfrac{6}{{12}} = \dfrac{1}{2}$
Therefore, the probability that neither will be alive in 10 more years is $\dfrac{1}{2}$.

Note: Given two events A and B which are mutually independent, i.e., the occurrence of one event does not depend on the occurrence of the other event, then $P(A \cap B) = P(A).P(B)$. In this question, the two events are mutually independent and so are the complements of the events. Therefore, $P(\overline M \cap \overline W ) = P(\overline M ).P(\overline W )$