Question

The first outcome is one of A and B the second outcome in each case is one of a and b. Use the following probabilities:

Pr[A] = [ 4/5]
Pr[B] = [ 1/5]
Pr[a | A] = [ 7/9]
Pr[b | A] = [ 2/9]
Pr[a | B] = [ 4/7]
Pr[b | B] = [ 3/7]
Pr[b] = [ 83/315 ]

What is Pr[B | b]?

Answer 1

I thought it would be Pr[B]/Pr[b], but it says that is wrong. So not 63/83

Answer 2

@DanielxAK @satellite73

Answer 3

Pr[B|b] = Pr[Bb]/Pr[b].

You don't know what Pr[Bb] is, since they don't give it to you. Can you think of another way to write Pr[Bb] in terms of what you know?

Answer 4

Pr[Bb] is (3/35)... I found Pr[Ba] to be 4/35 using Pr[a | B] = [ 4/7], so
4/35 + Bb=1/5, so Bb= (3/35)

Answer 5

Okay, so you should be able to solve it from there, correct?

Answer 6

That would be (1/5)/(3/35), but that comes out to 7/3, which it also says is wrong!

Answer 7

P[B|b] = P[Bb]/P[b]

Check what numbers you're plugging into your equation.

Answer 8

That worked! But why is it Pr[Bb]/Pr[b] and not Pr[B]/Pr[b]?

Answer 9

That's the definition. Given 2 events, A and B, you have:

P[A|B] = P[AB]/P[B]

P[AB] = P[A] only if B was a subset of A.

Answer 10

That explains a lot. Thank you!!