The probabilities of A and B dying in a year | vedclass

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

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$p + q - 2pq$

Vedclass · Answer

(B) Let $P(A) = p$ be the probability that $A$ dies,so $P(A') = 1 - p$ is the probability that $A$ lives.Let $P(B) = q$ be the probability that $B$ dies,so $P(B') = 1 - q$ is the probability that $B$ lives.The event that only one of them is alive at the end of the year means either ($A$ dies and $B$ lives) or ($B$ dies and $A$ lives).Since the events are independent,the required probability is $P(A \cap B') + P(B \cap A') = P(A) \cdot P(B') + P(B) \cdot P(A')$.Substituting the values: $p(1 - q) + q(1 - p) = p - pq + q - pq = p + q - 2pq$.

Outcome	Probability
$\omega_{1}$	$\frac{1}{14}$
$\omega_{2}$	$\frac{2}{14}$
$\omega_{3}$	$\frac{3}{14}$
$\omega_{4}$	$\frac{4}{14}$
$\omega_{5}$	$\frac{5}{14}$
$\omega_{6}$	$\frac{6}{14}$
$\omega_{7}$	$\frac{15}{14}$

The probabilities of $A$ and $B$ dying in a year are $p$ and $q$ respectively. What is the probability that only one of them will be alive at the end of the year?

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