Q.The probability that a man who is 45 years old will be alive till he becomes 70 is $\frac{5}{12}$. The probability that his wife who is 40 years old will be alive till she becomes 65 is $\frac{3}{8}$. What is the probability that, 25 years hence, a) the couple will be alive b) exactly one of them will be alive c) none of them will be alive d) at least one of them will be alive.

Find $P(A\cap B)$ from $P(A-B)=P(A)-P(A\cap B)$, then compare with $P(A)P(B)$.

Testing independence via $P(A\cap B)=P(A)P(B)$ is a core CBSE/NCERT Class 12 probability idea.

By definition $P(A-B)=P(A)-P(A\cap B)$, so

$P(A\cap B) = P(A) - P(A-B) = \frac14 - \frac16 = \frac{3-2}{12} = \frac{1}{12}.$

Now $P(A)\,P(B) = \frac14 \cdot \frac13 = \frac{1}{12}$.

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Exactly one of $A,B$ = (only $A$) + (only $B$) = $P(A)+P(B)-2P(A\cap B)$.

The probability of "exactly one" of two events is a CBSE/NCERT Class 12 probability result.

The event "exactly one of $A$ and $B$" is $(A \cap B') \cup (A' \cap B)$, whose probability is

$P(A\cap B') + P(A'\cap B) = [P(A)-P(A\cap B)] + [P(B)-P(A\cap B)].$

Combining:

$= P(A) + P(B) - 2P(A\cap B).$

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"Exactly one" $=P(A)+P(B)-2\,P(A\cap B)=x$ and "both" $=P(A\cap B)=y$; solve for $P(A)+P(B)$.

This reuses the "exactly one" identity from the CBSE/NCERT Class 12 probability chapter.

We are told:

$x = P(\text{exactly one}) = P(A)+P(B)-2P(A\cap B),$

$y = P(A\cap B).$

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$P(A/B)=P(A)$ and $P(B/A)=P(B)$ mean neither event's occurrence changes the other's probability — so $A$ and $B$ are independent events.

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