\problemdata                                                             
        {Probability} % latex-friendly label for the prob.
        {Probability} % The topic(s) of the problem content.  
        {from http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251/Site/} % Source (if known)                                          
        {S01 Final, 8} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}   
\begin{problemparts}        
\problempart What's the probability that $0$ doesn't appear
among $k$~digits chosen independently and uniformly at random?

\solution{ $9^k/10^k$.}

\problempart A box contains $90$ good and $10$ defective
screws. What's the probability that if we pick $10$~screws from the
box, none will be defective?

\solution{
$$\frac{\binom{90}{10}}{\binom{100}{10}} = \frac{(90!)^2}{100!80!}$$
}


\problempart First one digit is chosen uniformly at random from
$\{1,2,3,4,5\}$ and is removed from the set; then a second digit is
chosen uniformly at random from the remaining digits. What is the
probability that an odd digit is picked the second time?

\solution{ Condition on the event that the first digit is odd, get
  $\frac{2}{4} \cdot \frac{3}{5}+\frac{3}{4} \cdot \frac{2}{5} =
  \frac{12}{20} = \frac{3}{5}$.}

\problempart Suppose that you \emph{randomly} permute the
digits $1, 2,\cdots,n$, i.e., you select a permutation uniformly at
random. What is the probability the digit $k$ ends up in the
$i^{th}$ position after the permutation?

\solution{ $(n-1)!/ n! = 1/n$. This makes perfect sense, since it's a random permutation, so any digit ends up in position i with probability $\frac{1}{n}$.}


%From http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251/Site/
%Final exam review 1998

\problempart A fair coin is flipped $n$ times.  What's the
probability that all the heads occur at the end of the sequence?
\emph{Clarification:} If no heads occur, then ``all the heads are at
the end of the sequence'' (the statement is vacuously true).

\solution{$(n+1) {1/2}^{n}$.}


\end{problemparts}
\end{problem}
                                 