\documentclass{2800hw}
\usepackage{amsmath,amssymb,latexsym}
\usepackage{utf8}

\topic{Probability}
\homework{3}

\begin{document}
\maketitle

\begin{exercises}
\item \begin{enumerate}
  \item Suppose that $(S, P)$ is a probability space, and let $A$ be an event of
  $S$ with $P(A) ≠ 0$.  Define a function $Q$ by $Q(B) = P(B|A)$.  Prove that
  $(A,Q)$ is a probaility space.

  \item Suppose we define $R$ by $R(B) = P(A|B)$.  Give a counterexample to show
  that $(A,R)$ is not a probability space.
  \end{enumerate}

\item \begin{enumerate}
  \item Give a probability space and a pair of events $A$ and $B$ such that $A$
  and $B$ are mutually independent but are not mutually exclusive.

  \item Similarly, give an $A$ and $B$ that mutually exclusive but not mutually
  independent.

  \item Give an $A$ and $B$ that are neither mututally exclusive nor mutually
  independent.

  \item Give an $A$ and $B$ that are both mutually exclusive and mututally
  independent.
  \end{enumerate}

\item Show that for finite sample spaces, giving the probability of all of the
elementary events is sufficient to determine the probabilities of all events.
That is, if $S$ is finite, and if $(S, P_1)$ is a probability space, and if
$(S, P_2)$ is a probability space, and if for all $x \in S$, $P_1(\{x\}) =
P_2(\{x\})$, then for all $A \subseteq S$, $P_1(A) = P_2(A)$.

This is not true for infinite sample spaces; be sure to indicate in your proof
where you used the fact that $S$ is finite.

\item Sid and Mike go to the ice cream store. Sid likes chocolate ice cream,
but he doesn't want to order exactly the same flavour that Mike orders. If Mike
does not order chocolate ice cream, Sid will order it $90\%$ of the time. If
Mike does order chocolate ice cream, Sid will order it only $30\%$ of the time.
Mike chooses first. There are 5 different flavours of ice cream (only one is
chocolate), and Mike chooses a flavour completely at random (i.e. equiprobably).
If Sid ends up buying chocolate ice cream,
what is the probability Mike also ordered chocolate ice cream?

\end{exercises}
\end{document}
