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Question
The odds against a husband who is 60 years old, living till he is 85 are 7:5. The odds against his wife who is now 56, living till she is 81 are 5:3. Find the probability that exactly one of them will be alive 25 years hence
Solution
Let H ≡ the event that husband who is 60 is alive till age 85
W ≡ the event that wife who is 56 is alive till age 81
Odds against husband living till age 85 are 7:5
∴ P(H') = `7/(7 + 5) = 7/12`
∴ P(H) = 1 – P(H') = `5/12`
Odds against wife living till age 81 are 5:3
∴ P(W') = `5/(5 + 3) = 5/8`
∴ P(W) = 1 – P(W') = `3/8`
The required event will happen if husband is alive and wife is dead or husband is dead and wife is alive
∴ the required probability = P[(H ∩ W') ∪ (H' ∩ W)]
= P(H ∩ W') + P(H' ∩ W) ...[∵ events H ∩ W' and H' ∩ W are mutually exclusive]
= P(H) · P(W') + P(H') · P(W) ...[∵ events are independent]
= `5/12*5/8 + 7/12* 3/8`
= `46/96`
= `23/48`
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