\problemdata
{s01 ps10}
{random variable}

\begin{problem} Several statements involving probability
are listed below.  Assume that $A$ and $B$ are events, $S$ and $T$ are
random variables, and the underlying sample space is countable.
Furthermore, assume that all expectations exist.  For each statement,
describe restrictions on $A$, $B$, $S$, and $T$ sufficient to ensure
that the statement is true.  Make your restrictions as minimal as you
can.  No proofs are necessary.

\begin{problemparts}

\problempart $\Pr(A \cup B) = \Pr(A) + \Pr(B)$

\problempart $\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B)$

\problempart $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$

\problempart $\Pr(A \cap B) \leq \Pr(A)$

\problempart $\Pr(A \cap B) < \Pr(A)$

\problempart $\Ex(S + T) = \Ex(S) + \Ex(T)$

\problempart $\Ex(S \cdot T) = \Ex(S) \cdot \Ex(T)$

\end{problemparts}
\end{problem}