Question

If P(A) = 0.8, P(B) = 0.9, and P(A`nn`B)=.8, what is P(B|A)? What about P(A|B)?

Answer 1

`P(B|A) = 1`

`P(A|B) = 8/9 = 0.888dot`

Explanation:

If `P(A) = P(A|B)` then `AsubeB`
so given `B` it follows that `A` is certain.

One way to look at it is to suppose there were 1000 events
`P(A)=0.8` tells us that `800` of those events were of type `A`
`P(AnnB)=0.8` tells us that `800` of the `1000` are both `A` and `B`
So all the `A` events are "used up" in `(AnnB)`

Using the `1000` event example:
`P(B)=0.9` tells us `900` of the `1000` events are of type `B`

`800` of the `900` type `B` events are of type `A`,
so `P(A|B) = 800/900 = 8/9`