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.**