(Venn diagrams are helpful in understanding some of the events that arise below.)

(a) $\mathrm{P}(A \mid B)=\mathrm{P}(A \cap B) / \mathrm{P}(B)=\frac{1}{3}$

(b) $\mathrm{P}(B \mid A)=\mathrm{P}(A \cap B) / \mathrm{P}(A)=\frac{1}{5}$

(c) $\mathrm{P}(A \cup B)=\mathrm{P}(A)+\mathrm{P}(B)-\mathrm{P}(A \cap B)=0.7$, and the event $A \cap(A \cup B)=A$, so

$$

\mathrm{P}(A \mid A \cup B)=\mathrm{P}(A) / \mathrm{P}(A \cup B)=\frac{5}{7}

$$

(d) $\mathrm{P}(A \mid A \cap B)=\mathrm{P}(A \cap B) / \mathrm{P}(A \cap B)=1$, since $A \cap(A \cap B)=A \cap B$.

(e) $\mathrm{P}(A \cap B \mid A \cup B)=\mathrm{P}(A \cap B) / \mathrm{P}(A \cup B)=\frac{1}{7}$, since $A \cap B \cap(A \cup B)=A \cap B$.