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\pset{9}{November 15, 1994}{In tutorial, November 21 and 22}

{\bf Reading:\ }Maurer \& Ralston: Sections 6.1--6.3.

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\begin{problems}

\problem 
Consider the {\it Lets Make a Deal} game with the following assumptions
on how host Monty Hall hides the prize and on how you initially choose a door.
In each case, you follow the ``switch strategy'', and in each case Monty
reveals a loser after your initial choice.  State whatever assumptions
you need about how Monty reveals the loser.
For each case, determine the probability that you will win the prize by 
listing all events in the sample space 
along with the probabilities that they occur.
\bparts
\ppart
Monty Hall 
chooses a door to contain the prize with some non-uniform distribution---%
he hides the prize behind door number $i$ with some probability $p_i$, 
for $i=1,2,3$ and $\sum_i p_i =1$.
You, the contestant, (not knowing the $p_i$'s) follow the 
strategy of selecting a door
uniformly at random---you choose door $i$ with probability $1/3$.
\ppart
Monty Hall chooses a door behind which to 
hide the prize according to the uniform distribution 
but you the contestant select a door according to some non-uniform
distribution---you select door $i$ with some probability $p_i$, 
$\sum_{i=1}^3 p_i =1$.
\eparts

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\problem
Consider playing the {\it Lets Make a Deal} game with the following
additional information.
\bparts
\ppart
You are playing according to the original assumptions, but before
you make your initial choice, you happen to hear Keith (Carol) placing a 
``Nerd Pride'' pin underneath Box~1.  What should you do to try to win 
the prize (the ice cream)?  What is the probability that you win?
\ppart
Suppose Monty is determined to make you lose. 
In particular, if you initially choose a losing door, 
he may decide to end the game right there, opening your door to reveal
the goat you have ``won'', never giving you the chance to switch doors.
What is the probability of winning if you play the ``don't-switch strategy''?  
What if you play the ``switch strategy''?
\eparts

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\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?

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\problem
What is the probability of getting exactly 9 cards of the same suit
in a bridge hand (which consists of 13 randomly chosen cards)?
Define a ``5-card major'' to be a hand with either
exactly 5 spades or exactly 5 hearts or both.
What is the proabability of getting a 5-card major?

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\problem
In the lottery game Keno, a participant selects 20 different numbers from 
the set $S=\{1,\ldots, 80\}$.  
To determine winners, the lottery randomly selects 10 different numbers from 
$S$, and anyone who selected those 10 numbers (among his or her 20 numbers)
wins.

\begin{problemparts}
\problempart
What is the probability of winning Keno?
(You may leave factorials and powers in your answer.)
\remove{
  \problempart
  The game Teno is the played in the same manner as Keno, except that 
  the lottery selects 11 numbers instead of 10, and anyone who selects
  10 of those 11 (from among his or her 20 numbers) wins.
  What is the probability of winning Teno?
}
\problempart
Suppose each of the one-hundred students in 6.042 selects number to play
Keno.  We don't know how students will select their numbers; in particular,
they may not be selected uniformly or independently.
Give upper and lower bounds on the probability that there
is at least one winner in the class.
\ppart
Same as the last part, but say you know that exactly $k$ distinct 
tickets (sets of 20 numbers) were chosen, and that the $i$th
different ticket was chosen by exactly $p_i$ people.
\ppart
Same as part {\bf (b)}, but determine bounds on the probability
of there being {\em exactly \/} one winner.
\ppart
Same as the last part, but again exactly $k$ distinct 
tickets were chosen, and exactly $p_i$ people chose
the $i$th ticket.
\eparts

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\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 the criminal in this case has a beard?

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%%%	From Drake's book.
\problem	
To the best of our knowledge, with probability 0.8 Al is guilty of 
the crime for which he is about to be tried.
Bo and Cy, each of whom knows whether or not Al is guilty,
have been called to testify.
Bo is a friend of Al's and will tell the truth if Al is innocent
but will lie with probability 0.2 if Al is guilty.
Cy hates everybody but the judge and will tell the truth if Al is guilty
but will lie with probability 0.3 if Al is innocent.
\begin{problemparts}
\problempart
Determine the probability that the witnesses give conflicting testimony.
\problempart
Which witness is more likely to commit perjury?
\end{problemparts}


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\problem
A biased coin has probability $p$ of heads and probability $1-p$ of tails.
Suppose we flip this coin $n$ times.
\begin{problemparts}
\problempart
What is the probability of getting exactly $k$ heads?
\problempart
For what values of $k$ is the probability of getting exactly $k$ heads
greater than the probability of getting exactly $k-1$ heads?
For what values of $k$ is it less?
What value of $k$ is the most likely number of heads?
\problempart
Show that if $p=1/n$ then the probability of getting no heads is 
$\sim 1/e$ (for large $n$).
What is the ratio of the probability of no heads to the probability
of exactly one head?
\end{problemparts}


\end{problems}


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\end{problems}

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