Question

The probability that a man will be alive in $20$years is $\frac{3}{5}$ and the probability that his wife will be alive in $20$years is $\frac{2}{3}$. Then the probability that at least one will be alive in $20$years is

• A. $\frac{13}{15}$
• B. $\frac{7}{15}$
• C. $\frac{4}{15}$
• D. None of these

Answer 1

The correct option is A

$\frac{13}{15}$

The explanation for the correct answer.

Solve for the probability that at least one will be alive in $20$years.

$A$be the event that a man will be alive in $20$years

$B$be the event that his wife will be alive in $20$years.

Given that,$P\left(A\right)=\frac{3}{5},P\left(B\right)=\frac{2}{3}$

So,$P(A\text{'})=\frac{2}{5}$ and $P(B\text{'})=\frac{1}{3}$

The probability that at least one of them will be alive in $20$years$=1-$Probability that both $A$ and $B$ will die

Hence, the required probability $=1-\left[P\right(A\text{'})\times P(B\text{'}\left)\right]$

$$\begin{gathered} =1-\left[\left(\frac{2}{5}\right)\left(\frac{1}{3}\right)\right] \\ =1-\left(\frac{2}{15}\right) \\ =\frac{13}{15} \end{gathered}$$

Hence, option(A) is correct