Question

The odds against a man 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
(i)The couple will be alive 25 years hence
(ii)At least one of them will be alive 25 years hence.

Answer 1

First of all, we will calculate the probability of husband and wife living till 70 and 61 respectively using the formula: odds against the husband = $\dfrac{{{\text{P}}\left( {\overline H } \right)}}{{{\text{P}}\left( H \right)}}$and the odds against the woman: $\dfrac{{{\text{P}}\left( {\overline W } \right)}}{{{\text{P}}\left( W \right)}}$ . Then, P(H) = $\dfrac{5}{{5 + 7}} = \dfrac{5}{{12}}$ and we will find the other probabilities similarly.

Complete step-by-step answer:
We will solve the parts one by one. For the first part, we are asked the couple will be alive that means both are alive and it can be calculated using the formula: P (H$\cap$W) = P(H)P(W), where P(H) is the probability of the husband to be alive and P(W) is the probability of the wife to be alive.
For the second part, it is mentioned that at least one of the is alive after 25 years, that means we have to find the probability of any one of them being alive as well as the probability of both of them being alive. We will calculate it using the formula:
Probability = $P\left( H \right)P\left( {\overline W } \right) + P\left( {\overline H } \right)P\left( W \right) + P\left( H \right)P\left( W \right)$
Complete step by step solution: We are given the odds against a man who is now 45 years old, living till 70 as 7:5
The odds against the woman (wife of the man) who is now 36 years old, living till 61 as 5:3
Now, we know that odds against are given by the formula:
Odds against the husband = $\dfrac{{{\text{P}}\left( {\overline H } \right)}}{{{\text{P}}\left( H \right)}}$ = $\dfrac{7}{5}$ and the odds against the woman: $\dfrac{{{\text{P}}\left( {\overline W } \right)}}{{{\text{P}}\left( W \right)}}$= $\dfrac{5}{3}$
Now, the probability P($\overline {\text{H}}$) = $\dfrac{7}{{5 + 7}} = \dfrac{7}{{12}}$
Similarly, probability P (H) = $\dfrac{5}{{5 + 7}} = \dfrac{5}{{12}}$
Probability P ($\overline {\text{W}}$) = $\dfrac{5}{{3 + 5}} = \dfrac{5}{8}$
And, probability P (W) = $\dfrac{3}{{3 + 5}} = \dfrac{3}{8}$
Now, for part (I), we are asked the probability that the couple will be alive 25 years hence that means both, the husband and the wife, are alive.
We have the formula: P (H$\cap$W) = P(H)P(W), where P(H) is the probability of the husband to be alive and P(W) is the probability of the wife to be alive.
$\Rightarrow$ $P({\text{H}} \cap {\text{W}}){\text{ }} = {\text{ }}P\left( {\text{H}} \right)P\left( {\text{W}} \right)$= $\dfrac{5}{{12}} \times \dfrac{3}{8} = \dfrac{5}{{32}}$
Therefore, the probability of the couple being alive 25 years hence is found to be $\dfrac{5}{{32}}$ .
Now, for the part (II), we are asked the probability that at least one of them is alive that means either husband or wife is alive or both of them are alive 25 years alive.
This can be represented as: $P\left( H \right)P\left( {\overline W } \right) + P\left( {\overline H } \right)P\left( W \right) + P\left( H \right)P\left( W \right)$
$\Rightarrow$ Probability of at least one of them being alive = $\dfrac{5}{{12}} \times \dfrac{5}{8} + \dfrac{7}{{12}} \times \dfrac{3}{8} + \dfrac{5}{{12}} \times \dfrac{3}{8}$
$\Rightarrow$ Probability of at least one of them being alive = $\dfrac{{25}}{{96}} + \dfrac{{21}}{{96}} + \dfrac{{15}}{{96}} = \dfrac{{61}}{{96}}$
$\Rightarrow$ Probability of at least one of them being alive = $\dfrac{{61}}{{96}}$
Therefore, the probability of any one of them being alive is found to be $\dfrac{{61}}{{96}}$.

Note: In this question, you may get confused at many places such as finding the probability of the husband or wife being alive or not from the given odds against both of them. You may go wrong while solving the second part of this question because in that part, you need to calculate the probability of at least one of them being alive which will also include the probability of both of them being alive 25 years hence.