The probabilities 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: 

Required probability is 

$P\left[\right(A$ will die and $B$ alive) or $(B$ will die and $A$ alive)] 

$=P\left[\left(A\cap B^{\mathrm{\prime }}\right)\cup \left(B\cap A^{\mathrm{\prime }}\right)\right]$

Since events are independent, so 

Required probability $=P\left(A\right)\cdot P\left(B^{\mathrm{\prime }}\right)+P\left(B\right)\cdot P\left(A^{\mathrm{\prime }}\right)$

$=p\cdot (1-q)+q(1-p)=p+q-2pq.$