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\pset{11}{April 29, 1997}{May 6, 1997}


{\bf Reading:} Rosen 4.4-4.5 up to ``independence,'' 4.6.

In the problems below, try to simplify your answers as much as
possible.  There are a lot of questions, but many of them can be
answered with one equation and a brief explanation.

\begin{problems}


\problem  
Several dice and an urn.
\bparts
\ppart
When rolling a pair of fair dice, what is the probability of getting 
a sum of two on the dice? a sum of seven on the dice? a sum of eleven on 
the dice?

\ppart
Pepys wrote Newton to ask which of three events is more likely: that a
person
get (i) at least one six when $6$ dice are rolled, (ii) at least two sixes
when $12$ diced are rolled, or (iii) at least three sixes when $18$ dice are
rolled.  What is the answer?

\ppart %multinomial
A six-sided die is tossed 12 times.   What is the probability of
obtaining exactly 2 of each number from 1 to 6?
\ppart %combinations (easy)
An urn contains five balls numbered $1, 2, 3, 4, 5$.  Suppose
we draw three balls with replacement.  What is the probability that
all three balls are different?

\eparts



\problem % sample tree
When playing {\em Let's Make a Deal} in class, I decided to
``speed things up'' (give contestants a better chance of winning) by
adding a 4th box and, after your choice, revealing 2 that don't have
the prize.  Draw the tree of outcomes for the ``always switch'' rule
when 4 boxes are involved.  What is the probability that you win?

\problem % sample tree
Consider a best 3 out of 5 playoff.  If team 1 has a probability $1/2$
of winning each game that it plays, what is the most likely (that is,
highest probability) number of games in the playoff?  What if the
first team's win probability is $9/10$?

\problem %sample space, combinations, inclusion/exclusion
Suppose you are dealt a random poker hand of 5 cards from a
deck of 52 cards. Compute the probability that you are dealt
\bparts
\ppart Four of a kind (four cards with the same value)
\ppart A flush (5 cards from the same suit)
\ppart Two pair (two pairs, where each pair is two cards with the same
value, but the two pairs have different values and the remaining card
does not have either value)
\eparts

\problem % combinations
A batch of 50 items contains 10 defective items.  Suppose 5
of these items are selected at random (without replacement) and
tested.  
\bparts 

\ppart
What is the probability that exactly 4 of the items tested
are defective?  

\ppart What is the most likely number of defective items in the
sample?

\eparts


\problem %multinomial
Suppose that I randomly permute the letters in MISSISSIPPI.
What is the probability that all the S's end up next to each other?

\iffalse
\problem Suppose that I have $n$ dollars.  Each day I buy either a
pretzel (1\$), a soda (2\$) or a croissant (2\$).  
\bparts
\ppart
How many different
ways are there for me to spend my money?  Hint: devise and solve a
linear recurrence.
\ppart If all such ways are equally likely, what is the probability
that I never buy a pretzel?
\eparts
\fi

\problem Prove that if $A_1,\ldots,A_n$ is a partition of the sample space,
then
\[
\Pr[B] = \sum_{i=1}^n \Pr[B \mid A_i] \cdot \Pr[A_i]
\]

\problem

A witness to a crime says that the criminal had a beard.
We know from experience that if a criminal has a beard, 
witnesses will say so with probability~$0.8$, 
and if the criminal does not have a beard, witnesses will say that
he does with probability $0.12$.
If 15\% of the population has beards, what is the probability 
that a random criminal has a beard if the witness says he does?


\problemopt %sample tree
Suppose you are playing the {\em Let's Make a Deal} game, but
you happen to know that the prize is going to be behind doors 1, 2, or
3 with probabilities $p_1 \geq p_2 \geq p_3$, respectively.  How
should you play?  What are your chances of winning?  Can you prove
your strategy is optimal?


\problemopt
Let $f$ be a function $f\colon \{0,1\}^n \to \{0,1\}$.
We think of $f$ as an electoral system in which 
if people vote $X_1,\ldots, X_n$, 
$f(X_1,\ldots,X_n)$ wins the elections.

A natural requirement on $f$ is monotonicity:
\[\forall i.X_i\leq Y_i \Rightarrow
 f(X_1,\ldots, X_n)\leq f(Y_1,\ldots, Y_n),\]
i.e. if some people change their vote from 
$V$ to $1-V$, this can only change the 
result of the elections from $V$ to $1-V$.

We measure the quality of an  electoral system $f$
by the probability that a randomly selected 
person $R$ can change the outcome of the elections 
by changing his vote, when all other people vote 
independently and uniformly at random.

What is the the ``best" electoral system, i.e., 
which monotone function $f$ maximizes the 
probability that 
\[f(X_1,\ldots, X_R,\ldots, X_n)\neq 
  f(X_1,\ldots, 1-X_R,\ldots, X_n)\]
when $X_1,\ldots, X_n$ are independent 
uniform binary random variables 
and $R$ is an independent uniform random 
variable with values in $\{1,\ldots,n\}$?
 

\end{problems}

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