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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Discussion 1 -- Counting}
\centerline{September 14, 2009}
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{\it Background}
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We have discussed several techniques for counting 
different types of ``arrangements''
of elements from finite sets.  These are useful for
enumerating sample spaces in discrete probability problems.
The key facts:
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\item{1)}  (the ``$mn$ rule'') if $|A| = m$ and $|B| = n$,
then $|A\times B| = mn$.  (Hence, we can count
the number of elements in any set that can be put into 
one-to-one correspondence with the ordered pairs in $A\times B$.)
\item{2)}  If $|A| = n$, then the number of {\it ordered lists} 
(or ``permutations'') of size $r \le n$ chosen from $A$ is 
$$P_r^n = n(n-1)(n-2) \cdots (n-r+1).$$
\item{3)}  If $|A| = n$, then the number of {\it unordered lists}
(subsets, or ``combinations'') of size $r \le n$ that can 
be chosen from $A$ is 
$${n \choose r} = {n(n-1) \cdots (n-r+1)\over r!} = {n! \over r! (n-r)!}.$$
In today's discussion, we will practice applying these to 
several different types of counting problems.
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{\it Discussion Questions}
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{\it Note:} In your answers, explain carefully how you are using one
or more of the general patterns above to derive your results.
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A)  A standard deck of playing cards contains cards
identified by a {\it suit} (clubs, diamonds, hearts, spades) and 
{\it denomination} ($2,3,\ldots, 10$, jack, queen, king, ace).
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\item{1)}  Every possible combination of a suit and a denomination
occurs exactly once in the deck (e.g. there is exactly one 
10 of hearts).  How many cards are there in a complete deck?
\item{2)}  How many different 5-card hands can you be dealt in a 
game like poker (using a single standard deck)?  How many different
arrangements of cards are there if you are dealt a 5-card hand,
then lay the cards down left to right in front of you?  Explain.
\item{3)}  For the purposes of this question, a ``flush'' is a 
5-card hand in which all cards have the same suit (e.g. a hand of
5 clubs).  How many different hands are there of this type?  (If you know
what ``straight flushes'' and ``royal flushes'' are, note that they 
should be {\it included} in the hands you are counting here according 
to the definition of ``flush'' used here.)
\item{4)}  In basic poker, the weakest type of hand that is ``worth something''
is a hand containing exactly one pair of cards of the same
denomination (e.g. 2 8's), and three other cards, none of which
matches the denomination of the pair (or each other!). 
 How many different hands are there of this type?  
If you are dealt 5 cards randomly, what is the probability 
of getting one of these?

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B)  Students attending BigState University can choose from 
140 different major areas (Aardvark Husbandry to 
Zymurgy).  A student's major is identified in the Registrar's
computer data base by a two- or three-letter code (for 
example, the code for Cullinary Education is CE, while 
the code for Recreation Studies is RES).  Double majors have
just been approved by the faculty, and the Registrar wants
to know whether it is possible to assign a unique code as above 
for every possible combination of one or two majors.  
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\item{1)}  How many different possible double major programs are there?
How many programs with a single or a double major?
\item{2)}  How many different codes are 
there if all combinations of two or three capital letters are 
available?
\item{3)}  Is the current system of codes ``up to the job'' of 
accounting for double majors?
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C)  The rugby team is conducting a raffle to raise money to buy
new uniforms.  In all, 1000 \$1 tickets are sold, including 
25 tickets bought by the 25 team members (one apiece).
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\item{1)}  Suppose winners for three \$50 prizes are drawn 
randomly.  How likely is it that at least two prizes are won
by a team member?
\item{2)}  Would your answer change if the prizes 
were \$100, \$35, and \$15?  If so, how?  If not, why not?
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{\it Assignment}
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Group writeups due Monday, September 21.
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