Question

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

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

Answer 1

The correct option is B

$p+q-2pq$

Explanation for the correct option

Given that, the probability that A will die $=p$

the probability that B will die $=q$

The probability that only one of them will be alive $=P(A\cap B\text{'})+P(A\text{'}\cap B)$

As, the events are independent, the above form can be written as:

$$\begin{gathered} P(A\cap B\text{'})+P(A\text{'}\cap B)=P\left(A\right)·P(B\text{'})+P(A\text{'})·P\left(B\right) \\ P(A\cap B\text{'})+P(A\text{'}\cap B)=p(1-q)+(1-p)q \\ P(A\cap B\text{'})+P(A\text{'}\cap B)=p+q-2pq \end{gathered}$$

Hence the correct option is option(B) i.e. $p+q-2pq$