Finite Homework Help, Conditional Probability

Question

A and B are a subset of S, Pr[A|B]=4/9, Pr[A]=9/20, Pr[B']=11/20, what is Pr[B|A']?

Answer 1

This problem uses my personal favorite theorem, Bayes' theorem.

Bayes' Theorem states that for two random variables X and Y, P[X|Y] = P[Y|X] * P[X]/P[Y].

(This formula can be easily derived from the fact that P[X & Y] = P[X|Y] * P[Y] and also that P[X & Y] = P[Y|X] * P[X])

We want to use Bayes' theorem using B as X and A' as Y.

This tells us that

P[B|A'] = P[A'|B] * P[B]/P[A'].

We need to solve for each of P[A'|B], P[B], and P[A']

We are given that P[A|B] = 4/9. This means that P[A'|B] = 5/9.

We are given that P[B'] = 11/20, so P[B] = 9/20.

We are given that P[A] = 9/20, so P[A'] = 11/20.

We plug these into Bayes' theorem to get

P[B|A'] = P[A'|B] * P[B]/P[A']

P[B|A'] = 5/9 * (9/20) / (11/20)

P[B|A'] = 5/11.