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Question
The probability that a man who is 45 years old will be alive till he becomes 70 is `5/12`. The probability that his wife who is 40 years old will be alive till she becomes 65 is `3/8`. What is the probability that, 25 years hence,
- the couple will be alive
- exactly one of them will be alive
- none of them will be alive
- at least one of them will be alive
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Solution
Let A ≡ event that man who is 45 will be alive till age 70.
B ≡ event that wife who is 40 will be alive till age 65.
It is given that,
P(A) = `5/12`, P(B) = `3/8`
∴ P(A') = 1 – P(A) = `1 - 5/12 = 7/12`
∴ P(B') = 1 – P(B) = `1 - 3/8 = 5/8`
Since A and B are independent events,
A' and B' are also independent events.
(a) Let event C: Both man and his wife will be alive.
∴ P(C) = P(A ∩ B) = P(A) · P(B)
`= 5/12 xx 3/8`
`= 5/32`
(b) Let event D: Exactly one of them will be alive.
∴ P(D) = P(A' ∩ B) + P(A ∩ B')
= P(A') · P(B) + P(A) · P(B')
`= (7/12 xx 3/8) + (5/12 xx 5/8)`
`= 21/96 + 25/96`
`= 46/96 = 23/48`
(c) Let event E: None of them will be alive.
∴ P(E) = P(A' ∩ B') + P(A') · P(B')
`= 7/12 xx 5/8`
`= 35/96`
(d) Let event F: At least one of them will be alive.
∴ P(F) = 1 - P(none of them will be alive)
`= 1 - 35/96`
`= 61/96`
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