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\section*{CS2411 - Study Problems 1}
These problems are to review basic probability theory and to move you
along to Bayesian classification.  The example
class is an opportunity to get help or hints.
Problems marked with a \# are harder
or more mathematical. It is most important that you be able to do the
unmarked questions.

\subsection*{Simple Probability Calculus}

\begin{enumerate}
\item In craps (a game played with a pair of dice) you lose if you
  roll a 2 or an 11 on the first roll. What is the probability of
  losing on the first roll?
%\item  Two drunken students are playing Russian Roulette with
%  a revolver with six chambers. Fortunately, they are not aiming at
%  their heads. They are aiming at their computers. What is the
%  probability of one of the student's computer lasting six rounds of
%  the game? 
\item Two brothers skip school one morning and give the teacher the false
excuse that they had a flat tire. The teacher separates them and asks
them to each write down which tire was flat. What is the probability
that they both chose the same tire (assuming that they both guess)?
\item What is the probability of winning the jackpot on the National
Lottery? (To win the jackpot, one must match six different numbers
between 1 and 49 in any order).
\item Suppose in a given week the expected payoff in the National
Lottery is 8 million pounds (reduced slightly from the jackpot due to
possibility of shared prizes). Suppose a person buys a ticket for one
pound. Using probability as a model of belief, what can you say about that person's {\em belief} that they
will win as compared to the actual odds of winning? (Assume that the person behaves rationally and plays the
lottery solely to win. Are there other assumptions required to answer
this question?)
\item Suppose it is the case that 19\% of females smoke and 35\% have
blond hair. What is the probability that a randomly selected woman is
both a smoker and blond? What is the probability that she is either?
\item Make a histogram of the data in section~5.3 of the probability
  notes. What is your estimate of the probability of the value 35?

\end{enumerate}
\subsection*{Bayes Rule}

\begin{enumerate}
\item The English Speaking Traveller (from Pao). Suppose you are visiting a
country in which:
\begin{itemize}
        \item One out of 10 natives speaks English
        \item One out of five persons is a tourist 
        \item One out of two tourists speak English. 
\end{itemize}
You meet another person and converse in English. Would you deduce that
they are native? ({\em Use Bayes rule to compute 
the probability that they are native.})

\item In a city of a million inhabitants, a murder is committed. You
  are accused of the crime, although you are innocent. Some hair
is found at the scene which the genetic fingerprinting expert claims
matches your DNA, although you have never been there. During the
trial, the prosecutor asks this expert,
``what is the possibility of getting this same DNA signal from 
someone other than the defendant?'' The expert replies, ``the
probability of getting this DNA signal given that the hair is not 
from the defendant's head is one in a million.'' You turn to your
lawyer who makes the motion of a knife across the throat with his
finger. Express the information provided by the expert as a
  conditional probability. What is the conditional probability that you
  committed the crime, given his evidence? Use Bayes rule to compute
  it and get yourself acquitted.   
\end{enumerate}

\subsection*{From Probability to Statistics}
\begin{enumerate}
\item Suppose there is a process which produces identical computer
chips and it is your job to determine the reliability of the process.
You assume that the process produces a bad chip with probability $p$.
\begin{enumerate}
\item Suppose you know that $p$ is $0.1$. What is the probability 
that in a selected 10 chips, no bad chips will be found?
\item Suppose you test 10 chips and 5 are found to be bad. What 
is your estimate of $p$? How much would you expect the actual value 
of $p$ to vary from your estimate? What if you had tested 100 chips
and found 50 to be bad?
\item \# Suppose you test 10 chips and none are found to be bad. What 
can you conclude about $p$? In particular, how big can $p$ so that
this result is within a 95\% interval?
\end{enumerate} 
\end{enumerate}
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\subsection*{Bayes Classification}
\begin{enumerate}
\item Consider the following toy problem (from Beale and Jackson) -- you want
to classify people as either ballet dancers or rugby players based on their weight 
and height. You do some samples and find the results below:

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Because it appears that most of the variation is in the weight, you 
choose to use the weight as the only input feature for a classifier. 
Explain how you would make a Bayesian classifier from this data. That
is,
what conditional probabilities do you need for classification, what
conditional probabilities can you get from this data, and how would
you get it?  What 
would be the misclassification error on this data? (Don't
worry about the fact that the data is a bit silly).
\end{enumerate}
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