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\begin{document}

\centerline{
M331 \hfill
Spring 2012 \hfill
Final Exam \hfill 
A.Miller}

\probno=0 

\prob Suppose $x,y$ are two random reals in $[0,1]$ chosen
independently and uniformly.   What is the probability that $x^2$ is
less than $y$?

\answer{$\frac23$}

\prob
Suppose we have two red, two white, and two blue balls in a basket.

Choose 2, if they have the same color you win.

Choose 2 more (without replacing first two), if they
have the same color you win.

Finally if the two left in the basket are the same color, you
win.

\noindent What is the probability that you win?

\answer{$\frac7{15}$}

\prob
A box contains 3 cards:

One card has both sides white.

One card has both sides black.

One card has a white side and a black side.

\noindent A card is randomly drawn out
of the box and placed on a table.

Given that the side facing up is black
what is the probability that the other
side of the card is black?

\answer {$\frac23$}

\prob
Suppose $X$ is a nonnegative random variable
and $P(X<2)\leq \frac17$.  If $\mu=E(X)$
what is the smallest possible value $\mu$ can have?

\answer{$\frac{12}7$}

\prob
Let $X$ and $Y$ be random variables.
Prove that 

$E(XY)=E(X)E(Y)$ if and only if $\var(X+Y)=\var(X)+\var(Y)$.

\prob
In a long line of children you happen to notice that
about 20\% of the time a girl is followed by a boy and
about 70\% of the time a boy is followed by another boy.
If you randomly choose a child from the line, what
is the probability that its a girl?

\answer{$.6$}

\prob
Prove 
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$.

\prob
Suppose a basketball player has a probability of making
a 1-point shot of $\frac12$ and
a 2-point shot of  $\frac13$.
During a game he makes 12 1-point attempts and
27 2-point attempts.  Let
$X$ be the total number of points he scores.  Assuming
all shot attempts are pairwise independent Bernoulli trials,
find the expected value and variance of $X$.

\answer{24, 27}

\end{document}


