The probabilities that A and B will die withi | vedclass

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(B) Let $P(A)$ be the probability that $A$ dies $= p$,so $P(A') = 1 - p$ is the probability that $A$ is alive.Let $P(B)$ be the probability that $B$ d

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$p + q - 2pq$

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(B) Let $P(A)$ be the probability that $A$ dies $= p$,so $P(A') = 1 - p$ is the probability that $A$ is alive.Let $P(B)$ be the probability that $B$ dies $= q$,so $P(B') = 1 - q$ is the probability that $B$ is alive.We want the probability that only one of them is alive at the end of the year.This happens if ($A$ is alive and $B$ dies) or ($B$ is alive and $A$ dies).Required probability $= P(A' \cap B) + P(B' \cap A)$Since the events are independent,this is equal to $P(A')P(B) + P(B')P(A)$.$= (1 - p)q + (1 - q)p$$= q - pq + p - pq$$= p + q - 2pq$.

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

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