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\begin{document}

\pset{11}{November 19, 1996}{Start of Class, Tuesday, November 26, 1996}

{\bf Reading:} Lecture Notes on {\em Probability}.

\bigskip

\begin{problems}

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\problem {\bf [15 points]}

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 {\bf [20 points]}

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.)
\problempart
Suppose each of the 142 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 {\bf [15 points]}

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}

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\problem {\bf [10 points]}

Show that if two events $A$ and $B$ are independent, then the events $A$ 
and $\overline B$ are also independent.

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\problem {\bf [10 points]}

Show that if two random variables $R$ and $T$ are independent, then 
$R^2$ and $e^T$ are also independent.

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\problem {\bf [15 points]}

Use the formulas derived in class to compute upper bounds on the
following probabilities:
\bparts
\ppart 
In a poll with 2000 people, what is the probability that the result is off
by more than $1\%$?
\ppart
In 1000 independent coin flips, what is the probability of getting 
$\geq 600$ heads?
\ppart
In a noisy communication channel with $10\%$ error rate, what is the
probability that out of 1000 transmitted bits, $\geq 200$ bits are 
wrongly transmitted ({\em i.e.}, {\tt 1} is received as {\tt 0} and 
vice versa)?
\eparts

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\problem {\bf [15 points]}

Let $D_1$ and $D_2$ denote random variables for two independent dice 
experiments. Let $M_1=max(D_1,D_2)$, and $M_2=min(D_1,D_2)$ be two 
random variables.
Compute the distribution function for $M_1$. Are $M_1$ and $M_2$ 
independent random variables? Justify your answer.

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\end{problems}

\end{document}


