\problemdata     
{S01-PS8-4}           
{probability - intro}              
{} 		
{S99-PS9, S01-PS8}
{S01 Karger}





\begin{problem}


Consider the following procedure for
generating a small, positive integer.  I flip a coin.  If the outcome
is heads, then I roll one die and take the result.  If the outcome is
tails, then I roll two dice and take the sum.  Assume that the coin is
fair and the dice are fair and 6-sided.

\begin{problemparts}
\problempart Define a probability space that describes this experiment.

\solution{The sample space consists of 42 outcomes.  Six of these are
pairs:

\[
(H, 1), (H, 2), \ldots, (H, 6)
\]

Each of these outcomes occurs with probability $\frac{1}{2} \cdot
\frac{1}{6} = \frac{1}{12}$.  The remaining thirty-six outcomes are
triples:

\[
(T, 1, 1), (T, 1, 2), \ldots, (T, 6, 6)
\]

Each of these outcomes occurs with probability $\frac{1}{2} \cdot
\frac{1}{6} \cdot \frac{1}{6} = \frac{1}{72}$.}

\problempart What is the probability that the procedure generates a 5?

\solution{
\begin{eqnarray*}
\Pr(\text{generates 5})
        & = &   \Pr((H, 5)) + \Pr((T, 1, 4)) + \Pr((T, 2, 3)) \\
        &   &   \mbox{} + \Pr((T, 3, 2)) + \Pr((T, 4, 1)) \\
        & = &   \frac{1}{12} + \frac{1}{72} + \frac{1}{72}
                + \frac{1}{72} + \frac{1}{72} \\
        & = &   \frac{5}{36}
\end{eqnarray*}
}

\end{problemparts}

\end{problem}