Answer

1.

\[ \begin{aligned} P(A \cap B) &= P(A) P(B)\newline P(A) &= P((A \cap B) \cup (A \cap B^{c}))\newline &= P(A \cap B) + P(A \cap B^{c}) \quad\text{since disjoint}\newline P(A \cap B^{c}) &= P(A) - P(A)P(B)\newline &= P(A)(1 - P(B)) = P(A)P(B^{c}) \end{aligned} \]

2.

\[ \begin{aligned} (A \cup B)^{c} &= A^{c} \cap B^{c}\newline P(A^{c} \cap B^{c}) &= 1 - P(A \cup B)\newline &= 1 - P(A) - P(B) + P(A \cap B)\newline &= (1 - P(A))(1 - P(B))\newline &= P(A^{c})P(B^{c}) \end{aligned} \]