Question

the probability that a man will be alive in 25 years is 3/5 and probability for his wife in 25 year is 2/3. find probability that a) both will be alive in 25 years b) only the wife will be alive in 25 years solution

Answer 1

Task. The probability that a man will be alive in 25 years is $3/5$ and probability for his wife in 25 year is $2/3$. Find probability that:

a) both will be alive in 25 years;

b) only the wife will be alive in 25 years.

Solution. Let $X$ be the event that “a man will be alive in 25 years”, and $Y$ be the event that “his wife will be alive in 25 years”, so

$P(X)=3/5,\qquad P(Y)=2/3.$

Assume that these events are independent, so

$P(X\text{ and }Y)=P(X)*P(Y).$

a) We should find the probability that “both man and his wife will be alive in 25 years”, i.e. $P(XY)$. Since they are independent, we obtain that

$P(X\text{ and }Y)=P(X)*P(Y)=(3/5)*(2/3)=2/5.$

b) We should find the probability that “only the wife will be alive in 25 years”, i.e. $P(\bar{X}\text{ and }Y)$, where $\bar{X}$ is the opposite event to $X$, that is

$\bar{X}=\text{``a man will die before 25 years.''}$

Notice that

$P(\bar{X})=1-P(X)=1-3/5=2/5.$

Then

$P(\bar{X}\text{ and }Y)=P(\bar{X})\cdot P(Y)=(2/5)*(2/3)=4/15.$