\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}