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\title{MA162: Finite mathematics}
\subtitle{Combinatorial Probability}
\author{Paul Koester}
\institute{University of Kentucky}
\date{February 26, 2014}

\begin{document}

\maketitle

\begin{frame}[t]{Review of Empirical Probability}
The ``relative frequency'' approach to probability theory suggests defining the probability of an event as
\[ P(E) = \frac{n(E)}{n(S)}\]
where \(E\) is the event and \(S\) is the sample space.\\

Think ``Probability of event is number of favorable outcomes divided by total number of outcomes''

We've already used this ``relative frequency'' approach.  
The difference between today's lesson compared to previous lessons is that we will need to
rely on the more advanced counting techniques from Chapter 6.4.

\end{frame}

\begin{frame}[t]{Sock it to me!}
 \begin{itemize}
   \item Your dresser drawer has ten white socks, four black socks, and four plaid socks.  
   \item You reach in and pull out two socks.  What is the probability that you have a matching pair?
    \vfill
   \item You reach in and pull out three socks.  
    What is the probability that you can form a matching pair from these three socks?
    \vfill
  \end{itemize}
\end{frame}


\begin{frame}[t]{Drawing Cards}
 Two cards are drawn from a standard deck of 52 cards.
 (A recent HW question asked similar questions for a single card.  Notice the question for 2 cards is much more involved!)
 \begin{itemize}
  \item What is the probability the two cards have the same suit? \vfill
  \item What is the probability that a card has a number between 3 and 7? \vfill
  \item What is the probability that the two card hand contains a King or a Spade? \vfill
  \item What is the probability that the two card hand contains a King and a Spade?\vfill
 \end{itemize}
\end{frame}

\begin{frame}[t]{Poker Hands}
 \begin{itemize}
  \item A 5 card hand is drawn from a standard deck of 52 cards.
  \item Determine the probability of drawing a ``4 of a kind'' \vfill
  \item Now determine the probability of a ``Full house''  
   (3 of the cards have one rank and the other 2 cards have one rank)\vfill  
 \end{itemize}
\end{frame}

\begin{frame}[t]{Black Jack}
 \begin{itemize}
  \item Two cards are drawn from a standard deck of 52 cards.
  \item Determine the probability of being dealt a ``BlackJack.'' 
        (This means that one of the cards is an Ace and the other card has face value of 10.) \vfill
  \item Now suppose 2 decks are shuffled together, so that there are 104 cards.  
        What is the probability of being dealt a ``BlackJack'' from this larger deck.
 \end{itemize}
\end{frame}

\begin{frame}[t]{Urns}
 \begin{itemize}
  \item A bag contains 5 white balls, 12 red balls, and 9 green balls.
  \item You reach in and draw out 5 balls.
  \item What is the probability that 3 of the balls are red and 2 are green? \vfill
 \end{itemize}
\end{frame}

\begin{frame}[t]{Multiple Guess}
 \begin{itemize}
  \item An MA 123 exam consists of 20 questions, multiple choice
  \item There are 5 choices for each question
  \item A student randomly guesses on all of the questions.  
  \item What is probability student gets exactly 5 of the questions correct?  \vfill
  \item What is the probability this student gets at least an \(80\%\) on the exam?  \vfill
 \end{itemize}
\end{frame}

\begin{frame}[t]{Binomial Distribution}
\begin{itemize}
  \item The ``Multiple Guess'' problem is an example of a \alert{Binomial Probability}.
  \item An experiment is to be repeated n times.  
  \item At each stage, experiment may result in a ``success'' with probability \(p\) or ``failure'' with
        probability \(q = 1-p\).
  \item The outcome at any given stage does not influence the outcome at any other stage.
  \item The probability of exactly \(k\) successes is then 
  \[ \binom{n}{k} p^k q^{n-k} \]
\end{itemize}
\end{frame}

\begin{frame}[t]{Let the good times roll...}
 \begin{itemize}
  \item A fair six sided die is rolled 200 times.
  \item What is the probability of rolling a ``6'' exactly 40 times? \vfill
  \item What is the probability of rolling a ``6'' between 38 and 40 times?  \vfill
 \end{itemize}
\end{frame}

\begin{frame}[t]{Rollin', rollin', rollin'...}
 \begin{itemize}
  \item A pair of fair sided sided dice is rolled until you roll an 11.\\
  \item What is the probability that you need to roll the dice exactly 10 times? 
  (Thus, you roll an ``11'' on the 10th roll, and none of the first 9 rolls resulted in an ``11''.)
  \vfill 
 \end{itemize}
\end{frame}

\begin{frame}[t]{Geometric Distribution}
 \begin{itemize}
  \item The ``Rollin' '' problem is an example of a \alert{Geometric Probability}.
  \item At each stage, experiment may result in a ``success'' with probability \(p\) or ``failure'' with
        probability \(q = 1-p\).
  \item The outcome at any given stage does not influence the outcome at any other stage.
  \item You repeat the experiment until the first success.
  \item The probability that the first success occurs on the \(k^{\mbox{th}} \) trial is then 
  \[  p q^{k-1} \]
 \end{itemize}
\end{frame}

\begin{frame}[t]{Urns again}
 \begin{itemize}
  \item A bag of jelly beans contains 100 jelly beans.
  \item 32 of the jelly beans are red, your favorite flavor! 
  \item You reach in the bag and pick a jelly bean at random.  
  \item You keep doing this until you pick a red jelly bean.
  \item What is the probability that you will pick your first red jelly bean on the fifth pick?
 \end{itemize}
\end{frame}

\begin{frame}[t]{With or Without replacing?}
 \begin{itemize}
  \item There are two natural (and very different) approaches to this jelly bean question.
  \item Approach 1:  If you pick a non-red jelly bean then you put it back.  
        This is referred to as \alert{sampling with replacement}.
        At each stage, you are drawing from a bag of 100 jelly beans. 
        \vfill
  \item Approach 2:  If you pick a non-red jelly bean, then you put it aside (or throw it out, or whatever).
        This is referred to as \alert{sampling without replacement}.
        At each stage, you are drawing from a smaller bag, and so relative success increases slightly with each failure.
        \vfill
 \end{itemize}
\end{frame}

\end{document}
