Simple Probability Question - supersets

This seems like a relatively simple question, but it has been confusing me due to the specifics:

A baby born in 2016 has a 88% chance of living until the age of 64, and a 53.9% chance of living until age 84.

What is the probability that a baby born in 2016 will have a lifespan of between 64 to 84?

Obviously, living to 64 is a subset of living to 84, and I sketched out the venn diagram, but so far the answer I've got is

Let X be the lifespan of a baby:

$P (64 < X < 84) = 0.88 * (1-0.539) = 0.406$

This seems wrong to be as intuitively the probability should be greater than 0.539..

Any help is appreciated :)

• I think you mean that living to 84 is a subset of living to 64. Aug 19, 2017 at 15:14

Just think of it like there's three time windows. I.e.,

$P(t < T_1)$

$P(T_1 \le t < T_2)$

$P(t \ge T_2).$

Because those are the only three options and they're mutually exclusive, the probability of each of those three scenarios sum to one. You pretty much know the probabilities of the first and third scenarios.

• When you put it like that, It seems really stupid of me that I missed it. Thank you! :)
– Wboy
Aug 19, 2017 at 15:48