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%pset11 problem12 spring98; from fall 97 pset12
\begin{problem}
Give upper bounds for the following probabilities using the Markov,
Chebyshev, or Chernoff inequality, where appropriate.  If more than
one inequality applies, use the inequality that gives you the best
bounds. 

\begin{problemparts}
\problempart Suppose you play the following game:  Flip $5$ independent and
fair coins.  Score one point if the difference between the number of
heads and tails is at least $3$.  What is the probability that your
average score is greater than $0.9$ if you play the game $100$ times?

\problempart Let $a$ and $b$ be two numbers picked independently and
uniformly at random from the set $\{0,1,2,3,4\}$.
For $i = 0,1,2,3,4$, define the random
variable $X_i$ to be equal to $ai + b \bmod p$.  
Let $Y = \frac{1}{5}(X_1 + X_2 + X_3 + X_4 + X_5)$.  
What is the probability that $Y$ is greater than $3.5$?
{\em Hint}: Observe that $X_i$ is distributed uniformly on the set
$\{0,1,2,3,4\}$.  

\problempart You are told that the average (mean) annual income is \$10,000
in a particular population.  Five people are picked independently and
 uniformly and at random from the population.  
What is the probability that their avearge (mean) annual income is
greater than or equal to \$1,000,000?
\end{problemparts}
\end{problem}
%END   Ivan's problems
