Chapter 5, Problem 8CS

In Exercise 1-5, we computed the probability that a newborn life to a given age. Now let’s compute the probability that a person Aged x lives to age y. We’ll illustrate this with an example to compute the probability that a person age 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 (Alive at age 100|Alive at age 20)

Using the definition of conditional probability, we have $P\left(Aliveatage100|Aliveatage20\right)=\frac{P\left(Aliveatage100andAliveatage20\right)}{P\left(AIiveatage20\right)}$

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 alive at age 100 was also alive at age 20. Therefore,

P (Alive at age 100 and Alive at age 20) = P (alive at age 100)

Therefore, $P\left(Aliveatage100|Aliveatage20\right)=\frac{P\left(Aliveatage100\right)}{P\left(AIiveatage20\right)}$

In general, for y > x

$P\left(Aliveatage\text{ y}|Aliveatage\text{ x}\right)=\frac{P\left(Aliveatage\text{ y}\right)}{P\left(AIiveatage\text{ x}\right)}$

Which is more probable that a person aged 20 is still alive at age 50 or that a person aged 50 is still alive at age 60?