Question

If the probability that $A$ and $B$ will die within a year are $p$ and $q$ respectively, then the probability that only one of them will be alive at the end of the year is

• A. $p+q$
• B. $p+q-2pq$
• C. $p+q-pq$
• D. $p+q+pq$

Answer 1

The correct option is B $p+q-2pq$

Only one of $A$ and $B$ can be alive in the following, mutually exclusive ways.
(a) $A$ will die and $B$ will live
(b) $B$ will die and $A$ will live
So, required probability
$=P\left(a\right)+P\left(b\right)=p(1-q)+q(1-p)=p+q-2pq$