Question

The probabilities that Mr. $A$ and Mr. $B$ will die within a year are $\frac{1}{2}$ and $\frac{1}{3}$ respectively, then the probability that only one of them will be alive at the end of the year, is

• A. $\frac{5}{6}$
• B. $\frac{1}{2}$
• C. $\frac{2}{3}$
• D. None of the above

Answer 1

The correct option is B $\frac{1}{2}$

Let $A$ be the event that Mr. $A$ will die within a year and $B$ be the event that Mr. $B$ will die within a year. Then, required probability
$=P\left[\left(AwilldieandBalive\right)or\left(BwilldieandAalive\right)\right]$
$=P[(A\cap B^{{}^{\prime }})\cup (B\cap A^{{}^{\prime }})]$
$=P(A\cap B^{{}^{\prime }})+P(B\cap A^{{}^{\prime }})$ [$\because$ these events are mutually exclusive]
$=P\left(A\right)\cdot P\left(B^{{}^{\prime }}\right)+P\left(B\right)\cdot P\left(A^{{}^{\prime }}\right)$ [$\because A$ and $B$ are independent]
$=\frac{1}{2}(1-\frac{1}{3})+\frac{1}{3}(1-\frac{1}{2})$
$=\frac{2}{6}+\frac{1}{6}=\frac{3}{6}=\frac{1}{2}$