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\pset{9}{April 27,1995}{In recitation, May 5}

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\begin{problems}

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\problem
\points{10}
Suppose $A,B,C$ are independent events. Show that any of the combinations
\[ A^c, B,C; \, A, B^c,C; \cdots; A^c, B^c,C; \cdots; A^c,B^c,C^c \]
are also independent. Furthermore, show that $A$ and $B \cup C$ are 
independent.

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\problem

\points{10}
Chuck-A-Luck is played in many casinos. One version of it is as
follows.  There is a cage with three dice in it.  First the player
bets $ \$1$ on a number between $1$ and $6$.  The three dice are then
rolled.  If $i$ dice ($i=1,2$ or $3$) show his number, he wins $ \$i$
plus getting his initial $ \$1$ bet back; if none of the dice shows
the number, he loses his bet.

Define the expected value of some random variable $X$ to be 
$EX= \displaystyle \sum_{\mbox{all x}} x \cdot \mbox{Prob}\{X=x\}$. 

For a player starting with $\$1$, what is his expected wealth after
one play?

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\problem 

\points{10}

You are given a biased coin and are told that it could either be biased
towards heads (with $P[\text{H}]=\frac{2}{3}$) or biased towards tails (with
$P[\text{T}] = \frac{2}{3}$).

Your best knowledge is that it is equal likely to be either kind of coin.
So you flip the coin $12$ times and you get $7$ heads and $5$ tails. What is
your best guess about the type of coin and what is your estimate of the
probability that you are right?

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\problem

\points{10}

Once upon a time, a soldier was sentenced to death.  To show his kindness,
the king decided to give him another chance. He gave the soldier $50$ black
balls and $50$ white balls and $2$ pots and told him to divide the balls
between the two pots in whatever way he liked.  Then the king would come
back and randomly take away one pot and mix the balls in the other. The
soldier would then pick a ball from the remaining pot and live if and only
if it was white. What strategy should the soldier take? Prove that it is the
best.

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\problem 
\points{10}

Consider in a country where people favor boys over girls and each family
will keep having babies until they have a boy. (And then they stop
having kids.)  Assume that genetically boys and girls are equally
likely to be born, and that the death rate is the same for both sexes.
Will there be a bias in that country's population in the long run?
Explain your answer.
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\problem 
\points{10}
Mr. James W. Pitman received a letter: \\
`` $\cdots$ to inform you that one of the following three people has
 won a New Datsun Sentra: Collin Andrus; James W. Pitman; Larry Abbott. 
$\cdots$"\\
`` $\cdots$ you will receive one of the following for sure:\\
$(1)$ RCA Color TV; $(2)$ 5 FT. Grandfather Clock; $(3)$ Datsun Nissan sentra.\\
To claim your gift, call $\cdots$"\\
According to the small print on the back of the letter:\\
`` $\cdots$ the retail values and chance of receiving each gift are No. 1--
$1/10,000$ ($ \$329.95$), No.2-- $9998/10,000$ ($ \$249.95$), No.3--
$1/10,000$ ($ \$5,995.00$). $\cdots$ "\\
Let us assume that this is an honestly conducted sweepstake, and each of the
three individuals named above had originally a $1$ in $10,000$ chance of 
winning the new Datsun. Now that Mr. Pitman knows that the winner is one of
three individuals (and he is on the list), the rules of conditional probability
imply that he has a $\frac{1}{3}$ chance of winning the Datsun. True or
False? Explain!

\end{problems}

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