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\centerline{MONT105Q -- Mathematical Journeys}
\centerline{Problem Set 2}
\centerline{{\bf due:} Friday, March 4, 2016}
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\noindent
I.  Four cards will be dealt off the top of a well-shuffled deck (without replacement,
of course!). 
\begin{enumerate}
\item[(A)]   Which is more likely, or are they equally likely?  Explain and/or 
calculate probabilities:
\begin{enumerate}
\item[(i)] The cards are a spade, then a heart, then a diamond, then a club, or
\item[(ii)] the cards belong to four different suits?
\end{enumerate}
\item[(B)] Again, which is more likely and explain:
\begin{enumerate}
\item[(i)] The cards all belong to the same suit, or
\item[(ii)] the cards belong to four different suits?
\end{enumerate}
\end{enumerate}
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\noindent
II.  One ticket will be drawn at random from a box containing 6 tickets: 
three with the numbers 2,2,3 on blue paper, and three with the numbers 2,2,3 on green 
paper.  Are the number and the color independent?  Why or why not?
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\noindent
III.  A fair, standard die is rolled 10 times.  Thinking of the results
as \emph{ordered} sequences of numbers 1,2,3,4,5,6, find the probability of 
\begin{enumerate}
\item[(A)] getting 10 sixes
\item[(B)] not getting 10 sixes (that is, anothing other than 10 sixes)
\item[(C)] all rolls showing four or fewer spots
\item[(D)] the sum of the numbers shown in the 10 rolls is greater than or equal
to 57.
\end{enumerate}
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\noindent
IV.  The probability of an event $A$ is $1/4$ and the probability of 
a second event $B$ is $1/5$. True or false, and explain:
\begin{enumerate}
\item[(A)] If $A$ and $B$ are independent, they must be mutually exclusive.
\item[(B)] If $A$ and $B$ are mutually exclusive, then they cannot be independent.
\item[(C)] If the probability of $A {\rm or} B$ is $9/20$, then $A$ and $B$
are mutually exclusive.
\item[(D)] If $A$ and $B$ are independent, then the conditional probability 
$P(A|B)$ is $1/4$.   
\end{enumerate}
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\noindent
V.  A fair coin will be tossed 10 times.  What is the probability that exactly
2 of the first 5 tosses will yield heads, and exactly 3 of the last 5 tosses
will yield heads?  What are you assuming here?  Also, \emph{without calculations,
explain} whether your answer is less than, greater than, or equal to the probability
that there are exactly 5 heads in the 10 tosses.  
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\noindent
VI.  A (very) simple keno game works like this:  You make a $\$ 1$ on a single number.
If your number comes up you get your bet back and $\$ 2$ more.  If not, you 
lose your bet.  You have a 1 in 4 chance of matching your number.  
\begin{enumerate}
\item[(A)]  State a box model that is equivalent to the outcome of this game.
(That is, the tickets should tell the dollar amounts you win or lose and 
a draw from the box should be the same as playing one round of the game.)
\item[(B)] What is the
expected value of the amount you win (or lose) in one play?
\item[(C)] How much would you expect to win or lose if you played $100$ times?
\end{enumerate} 
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