Chapter 5, Problem 9CS

In Exercises 1-5, we computed the probability that a newborn lives to a given age. Now let’s compute the probability that a person aged x fives to age y. We'll illustrate this with an example to compute the probability that a person aged 20 lives to age 100. This is the conditional probability that a person lives to age 100, given that the person has lived to age 20.

We want to compute the conditional probability $P\left(\text{Alive at age 100 | Alive at age 20}\right)$

Using the definition of probability, we have $P\left(\text{Alive at age 100 | Alive at age 20}\right)=\frac{P\left(\text{Alive at age 100 | Alive at age 20}\right)}{P\left(\text{Alive at age 100}\right)}$

You computed P (Alive at age 20) in Exercise 2. Now we need to compute P(Alive at age 100 and Alive at age 20). The key is to realize that anyone who is awe at age 100 was also alive at age 20. Therefore, $P\left(\text{Alive at the age 100 and Alive at age 20}\right)=P\left(\text{Alive at the age 100}\right)$

Therefore, $P\left(\text{Alive at the age 100 and Alive at age 20}\right)=\frac{P\left(\text{Alive at the age 100}\right)}{P\left(\text{Alive at the age 20}\right)}$

In general: for $y>x,$ $P\left(\text{Alive at the age y | Alive at age x}\right)=\frac{P\left(\text{Alive at the age y}\right)}{P\left(\text{Alive at the age x}\right)}$

A life insurance company sells term insurance policies These policies pay $100,000 if the policyholder dies before age 70, but pay nothing if a person is still alive at age 70. If a person buys a policy at age 40, what is the probability that the insurance company does not have to pay?