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\begin{document}
\handout{tut10}{Tutorial 10 Problems}{April 17, 1996}
\begin{problems}

\problem 
Suppose you are playing the {\it Lets Make a Deal} game with
4 doors rather than~3.  Say Monty and you choose doors according to
the uniform distribution.  After your initial choice, Monty reveals a
loser and offers you the chance to switch.  Subsequently, he reveals
another loser and again offers you the chance to switch.  Compute the
probability of winning for each of the four possible strategies: (1)
stick both times, (2) switch both times, (3) stick the first time,
switch the second time, and (4) switch the first time, stick the
second time.  Which strategy is the best?

\problem
The IRS has decided to audit $k$ tax returns from the list of those
submitted.  It doesn't know how exactly how many returns it has
(although it has many more than $k$), and it also wants to be fair so
that each tax form is equally likely to be audited.
Thus, the IRS will perform a one-time pass over the tax returns and
collect $k$ returns to be auditted.
During this pass, the IRS will visit each return, and it has the option
to add that return to the audit list.   However, it can hold 
at most $k$ returns at a single time, so the IRS may also
remove returns from its audit list.

Devise a randomized strategy for the IRS, so that at any given time, the
probability of every tax return (from the set of returns that have
been examined) is equally
likely to be in the audit list.  After examining $k$ returns, the
audit list must contain $k$ returns.

{\em Hint:  Consider $k=1$, and then try to generalize.}

\end{problems}
\end{document}
