\documentclass{2800hw}
\usepackage{amsmath,amssymb,latexsym}
\usepackage{enumitem}

\begin{document}

\name{Siddhartha Chaudhuri}
\netid{sc2663}
\homework{1}

\maketitle

\begin{exercises}

\item
{\bf (3 points)} {\bf A broken proof:} A student is asked to prove that 17 = 35. They submit the following ``proof'':
\[
\begin{array}{rcll}
  17                                  & = & 35 \\
  17 - 26                             & = & 35 - 26                             & \textrm{(subtract 26 from both sides)} \\
  (17 - 26)^2                         & = & (35 - 26)^2                         & \textrm{(square both sides)} \\
  17^2 - 2 \times 26 \times 17 + 26^2 & = & 35^2 - 2 \times 26 \times 35 + 26^2 & \textrm{(expand)} \\
  17^2 - 2 \times 26 \times 17        & = & 35^2 - 2 \times 26 \times 35        & \textrm{(cancel $26^2$)} \\
  17^2                                & = & 35^2 - 2 \times 26 \times 18        & \textrm{(add $2 \times 26 \times 17$ to both sides)} \\
  289                                 & = & 1225 - 936                          & \textrm{(plug it into a calculator)} \\
  289                                 & = & 289                                 & \textrm{(Q.E.D.)}
\end{array}
\]
Explain the faulty reasoning in this ``proof''.

\item Prove from first principles (set theory, Kolmogorov's axioms) or give a counterexample for each of the following:

\begin{enumerate}[label=\alph*)]
\item {\bf (3 points)} $P(A \cap B) \ \leq \ P(A)$ for any events $A$ and $B$.
\item {\bf (3 points)} If $A \subseteq B$ but $A \neq B$ then $P(A) < P(B)$.
\item {\bf (3 points)} If $A_1, A_2, \dots, A_n$ are mutually exclusive events and $\bigcup_i A_i = S$ (the entire sample space), then for some $i$, $P(A_i) \geq 1/n$.
\end{enumerate}

\item
{\bf (3 points)} Two Cornell students missed their final exam because they were partying in New York City the night before. Desperate for a make-up test, they lied to the professor that they had a flat tire while returning. The professor agreed to give them a make-up test, as long as the students sat in separate rooms. When they opened the paper, they found a single question, worth 100 points: ``Which tire was it?''

What's the probability the two students will give the same answer? Justify your result and clearly state any assumptions you made.

\item {\bf Drawing pairs:}
\begin{enumerate}[label=\alph*)]
\item {\bf (2 points)} You have 10 red, 10 green and 10 blue pairs of socks in a drawer. What's the probability that if you randomly pull out two socks without looking, they will be the same color?
\item {\bf (3 points)} You have 10 red, 10 green and 10 blue pairs of shoes in a (very large) drawer. What's the probability that if you randomly pull out two shoes without looking, they will be the same color {\em and} a left-right pair?
\end{enumerate}
Justify your answer in both cases.

\end{exercises}

\end{document}
