The odds against a husband who is 45 years old, living till he is 70 are 7:5 and the odds - Maths - Probability

Question

The odds against a husband who is 45 years old, living
till he is 70 are
7:5 and the odds against his wife who is now 36, living till
she is 61
are 5:3 Find the probability that a) the couple will alive 25
years hence
b) exactly one of them will be alive 25 years hence c) none
of them will
be alive 25 years hence d) at least one of them will be alive
25 years
hence

Ajanta Trivedi · Accepted Answer

the odds against a husband will alive 25 years hence = 7:5

the probability that a husband will alive 25 years hence = $\frac{5}{7 + 5} = \frac{5}{12}$

the probability that husband will not be alive = $\frac{7}{12}$

the odds against his wife will alive 25 years hence = 5:3

the probability that a wife will alive 25 years hence = $\frac{3}{5 + 3} = \frac{3}{8}$

the probability that wife will not be alive = $\frac{5}{8}$

(a)

the probability that both will alive = $\frac{5}{12} * \frac{3}{8} = \frac{5}{32}$

(b)

exactly one of them will alive i.e. either husband will alive or wife will alive

the probability that husband will be alive and wife will not be alive = $\frac{5}{12} * \frac{5}{8} = \frac{25}{96}$

the probability that wife will be alive and husband will not be alive = $\frac{3}{8} * \frac{7}{12} = \frac{21}{96}$

the probability that exactly one of them will be alive = $\frac{25}{96} + \frac{21}{96} = \frac{25 + 21}{96} = \frac{46}{96} = \frac{23}{48}$

(c)

probability that none of them will be alive = $\frac{7}{12} * \frac{5}{8} = \frac{35}{96}$

(d)

probability that at least one of them will be alive = 1- probability that none of them will be alive

$= 1 - \frac{35}{96} = \frac{96 - 35}{96} = \frac{61}{96}$

hope this helps you.