Chapter 11 - Counting Methods and Probability Theory - Chapter Summary, Review, and Test - Review Exercises - Page 761: 91

See below.

a) We know that $P(\text{A and B})=P(A)P(B|A)$, thus $P(B|A)=\frac{P(\text{A and B})}{P(A)}=\frac{\frac{2}{9}}{\frac{4}{9}}=\frac{1}{2}$ b) We know that $P(\text{A and B})=P(A)P(B|A)$, thus $P(B|A)=\frac{P(\text{A and B})}{P(A)}=\frac{\frac{2}{9}}{\frac{7}{9}}=\frac{2}{7}$