\setcounter{ExampleCounter}{1}
You probably have some idea of what we mean when we say ``probability,'' but here's a definition to clarify:
\begin{proc}{What is Probability?}
\paragraph{Probability:} a way of describing how certain we are of the result of a particular experiment or activity.
\paragraph{Probability of something occurring:} it is defined as the proportion of times that it would occur if we repeated the experiment over and over.
\end{proc}
\subsection{Vocabulary}
\begin{itemize}
\item \textbf{Outcome:} One possible result of an experiment.\\
Ex: flipping a coin is an experiment; heads is one outcome, and tails is another
\item \textbf{Sample Space:} The list of all possible outcomes. We can express this in several ways:
\begin{enumerate}[(a)]
\item List the outcomes in set notation.\\
Ex: rolling a six sided die: \[S = \left\{1, 2, 3, 4, 5, 6 \right\}\]
\item Create a tree diagram, showing different ways that events in order could happen.
\item Draw a Venn diagram (we'll see this later in the chapter).
\end{enumerate}
\vfill
\pagebreak
\item \textbf{Event:} one or more outcomes.\\
Ex: rolling a six sided die:
\begin{align*}
A &= \textrm{ rolling a } 4\\
B &= \textrm{ rolling an odd number}\\
C &= \textrm{ rolling a number greater than } 2
\end{align*}
The probability of an event $A$ is written $P(A)$, or we could write $P($rolling a $4)$ or $P(4)$, if that is clear enough in context.
\end{itemize}
\begin{example}{Two siblings}
Consider randomly selecting a family with 2 children where the order in which different gender siblings are born is significant. That is, a family with a younger girl and an older boy is different from a family with an older girl and a younger boy. What would the sample space look like? \\
\marginnote{\bfseries Solution} If we let G denote a girl, B denote a boy, then we have the following:
\[ S = \left\{ GG, BB, GB, BG \right\} \]
This notation represents families with 2 girls, 2 boys, an older girl and a younger boy, an older boy and a younger girl.
\end{example}
\begin{example}{Three Siblings}
What would the sample space $S$ look like if we considered a family with 3 children? Remember, the order of children born is significant. \\
Now there are 8 possibilities:
\[S = \{GGG, BBB, GGB, GBG, BGG, BBG, BGB, GBB\}\]
\end{example}
The following example describes a familiar experiment that can actually be easily performed.
\begin{example}{Tossing a coin and rolling a die}
Suppose we toss a fair coin and then roll a six-sided die once. Describe the sample space $S$. \\
\marginnote{\bfseries Solution \\ \includegraphics[height = .3in]{dice}} Let $T$ denote Tails, and $H$ denote Heads. Then:
\[ S = \left\{T 1, T 2, T 3, T 4, T 5, T 6, H 1, H 2, H 3, H 4, H 5, H 6 \right\} \]
\end{example}
\vfill
\pagebreak
\paragraph{Note:} in the definition of probability, we said that probability is a \textbf{proportion}.\\
\begin{proc}{Probability}
The basic rules of probability are:
\begin{enumerate}
\item $0 \leq P(A) \leq 1$ for any event $A$; that is, all probabilities are between 0 and 1
\item $P(A) = 0$ means that event A will not occur
\item $P(A) = 1$ means that event A is certain to occur
\item $P(E_1) + P(E_2) + \dots + P(E_n) = 1 $; that is, the sum of probabilities of all possible outcomes of an experiment $E$ is 1
\end{enumerate}
\end{proc}
Often we use percentages to represent probabilities. For example, a weather forecast might say that there is 85\% chance of rain in Frederick tomorrow. Or there is 67\% chance that the Baltimore Orioles will win their next series. Or a particular poker player has a 35\% chance of winning the game with his current hand. As you might have already guessed, 100\% chance corresponds to 1, and 0\% corresponds to 0.
\subsection{Theoretical Probability}
There are two types of probability: \textbf{theoretical} and \textbf{empirical}. Theoretical probability is used when the set of all equally-likely outcomes is known. To compute the theoretical probability of an event $A$, denoted $P(A)$, we use the formula below:
\begin{formula}{Theoretical probability}
\[ P(A) = \dfrac{\textrm{number of ways } A \textrm{ can occur}}{\textrm{total number of possible outcomes}} \]
\end{formula}
This makes sense with the definition of probability, namely that it is the proportion of times we would expect $E$ to occur if we repeated the experiment many times. This proportion comes from dividing the number of possibilities that correspond to $E$ by the total number of possibilities there are.
\vfill
\pagebreak
In the example below, one could probably find the probability by intuition, but it's good to know how to apply the formula, even in what seems to be a simple experiment.
\begin{example}{Rolling a die}
Assume you are rolling a fair six-sided die. What is the probability of rolling an odd number?\\
\marginnote{\bfseries Solution} \emph{Since half of the sides of a die have an even number of pips, and the other half are odd, intuitively you know that there is 50\% chance of rolling an odd number. But how would you compute this probability formally?} \\
There are 6 possible outcomes when rolling a die: 1, 2, 3, 4, 5, and 6. Three of these outcomes are odd numbers: 1, 3, and 5. Let $O$ denote an event when an odd number is rolled. Then
\[ P(O) = \frac{3}{6} = \frac{1}{2} \]
\end{example}
\begin{example}{Three Siblings}
Consider the earlier example about a family with three children. Remember, the sample space looked like
\[S = \{GGG, BBB, GGB, GBG, BGG, BBG, BGB, GBB\}.\]
Let's find the probability of a few combinations of kids:
\begin{enumerate}[(a)]
\item $P($three girls$) = \dfrac{1}{8}$
\item $P($at least two boys$) = \dfrac{4}{8}$
\item $P($exactly one girl$) = \dfrac{3}{8}$
\item $P($youngest is a boy$) = \dfrac{4}{8}$ (note why this makes sense; independent trials)
\item $P($oldest and youngest same$) = \dfrac{4}{8}$ (note why this also makes sense)
\end{enumerate}
\end{example}
\vfill
\pagebreak
In the next example, it is not necessary to list all possible outcomes of an experiment. However, if you are not familiar with a standard deck of 52 cards, the diagram below should be helpful.
\begin{center}
\includegraphics[width=0.8\textwidth]{playingcards}
\end{center}
\begin{example}{Drawing a card}
Suppose you draw one card from a standard 52-card deck. What is the probability of drawing an Ace? \\
\marginnote{\bfseries Solution} There are 4 aces in a deck of cards. Let $A$ denote an event that the drawn card is an Ace. Then
\[ P(A) = \frac{4}{52} = \frac{1}{13} \]
\end{example}
\begin{example}{Drawing Another Card}
\begin{enumerate}[(a)]
\item When drawing a card from a standard 52-card deck, what is the probability of drawing a face card? \emph{Face cards include Jacks, Queens, and Kings}.
\[\dfrac{12}{52}\]
\item What is the probability of drawing the King of Hearts?
\[\dfrac{1}{52}\]
\end{enumerate}
\end{example}
\vfill
\pagebreak
\begin{example}{Cookie Jar}
Lisa's cookie jar contains the following: 5 peanut butter, 10 oatmeal raisin, 12 chocolate chip, and 8 sugar cookies. If Lisa selects one cookie, what is the probability she gets a peanut butter cookie? \\
\marginnote{\includegraphics[height = .8in]{cookies} } The total number of cookies in the jar is 35. Let $PB$ denote the event when a peanut butter cookie is selected, then
\[ P(PB) = \frac{5}{35} = \frac{1}{7} \]
\end{example}
\subsection{Empirical Probability}
As long as we can list--or at least count--the sample space and the number of outcomes that correspond to our event, we can calculate basic probabilities by dividing, as we have done so far. But there are many situations where this isn't feasible.
For instance, take the example of a batter coming to the plate in a baseball game. There's no way to even begin to list all the possible outcomes that could occur, much less count how many of them correspond to the batter getting a hit. We'd still like to be able to estimate the likelihood of the batter getting a hit during this at-bat, though. Just as sports fan do, then, we turn to this batter's previous performance; if he's gotten a hit in 200 of his last 1000 at-bats, we assume that the probability of a hit this time is $\frac{200}{1000}=0.200$.
Empirical probability is used when we observe the number of occurrences of an event. It is used to calculate probabilities based on the \emph{real data} that we observed and collected. To compute the empirical probability of an event $E$, denoted $P(E)$, we use the formula below:
\begin{formula}{Empirical probability}
\[ P(E) = \dfrac{\mbox{observed number of times $E$ occurs}}{\mbox{total number of observed occurrences}} \]
\end{formula}
\vfill
\pagebreak
This can also be used to answer questions about sampling randomly from a population if we know the breakdown of the group.
\begin{example}{FCC students}
Consider the following information about FCC students' enrollment:
\begin{center}
\begin{tabular}{|c|c|}
\hline
Gender & Enrollment \\ \hline
Female & 3653 \\ \hline
Male & 2580 \\ \hline
\end{tabular}
\end{center}
If one person is randomly selected from all students at FCC, what is the probability of selecting a male?\\
\marginnote{\bfseries Solution} The total enrollment is 6233 students, thus we get:
\[ P(M) = \frac{2580}{6233} \approx 0.414 \]
\end{example}
The next example contains a two-way table, often referred to as \emph{contingency} table, which breaks down information about a group based on two criteria. For example, the table below breaks down a group of 130 FCC students based on gender and which hand is their dominant hand:
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Gender & Right-handed & Left-handed \\ \hline
Female & 58 & 13\\ \hline
Male & 47 & 12 \\ \hline
\end{tabular}
\end{center}
In order to use this to calculate probabilities if we randomly select someone from the group, we need to calculate totals for each category: the number of males, the number of females, the number of left-handed people, and the number of right-handed people. This is done by simply summing each row and column; if we do that, we obtain the completed table below.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Gender & Right-handed & Left-handed & \textbf{Total} \\ \hline
Female & 58 & 13 & 71\\ \hline
Male & 47 & 12 & 59 \\ \hline
\textbf{Total} & 105 & 25 & 130 \\ \hline
\end{tabular}
\end{center}
\vfill
\pagebreak
\begin{example}{FCC students}
Consider the following information about a group of 130 FCC students:
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Gender & Right-handed & Left-handed & \textbf{Total} \\ \hline
Female & 58 & 13 & 71\\ \hline
Male & 47 & 12 & 59 \\ \hline
\textbf{Total} & 105 & 25 & 130 \\ \hline
\end{tabular}
\end{center}
\begin{enumerate}[(a)]
\item If one person is randomly selected from the group, what is the probability this student is left-handed? \\
The total number of left-handed students in the group is 25, thus \[ P(L) = \frac{25}{130} \approx 0.192 \]
\item Find the probability of selecting a female student.\\
\[P(F) = \dfrac{71}{130} \approx 0.546\]
\end{enumerate}
\end{example}
\vfill
\pagebreak
\begin{comment}
\begin{exercises}
\pthree{A fair die is rolled. Find the probability of getting 4.}
\pthree{A fair die is rolled. Find the probability of getting less than 3.}
\pthree{A fair die is rolled. Find the probability of getting at least 5.}
\ptwo{You have a bag with 20 cherries, 14 sweet and 6 sour. If you pick a cherry at random, what is the probability that it will be sweet?}
\ptwo{A ball is drawn randomly from a jar that contains 6 red balls, 2 white balls, and 5 yellow balls. Find the probability of drawing a white ball.}
\ptwo{Suppose you write each letter of the alphabet on a different slip of paper and put the slips into a hat. What is the probability of drawing one slip of paper from the hat at random and getting a consonant?}
\ptwo{In a survey, 205 people indicated they prefer cats, 160 indicated they prefer dogs, and 40 indicated they don't enjoy either pet. Find the probability that if a person is chosen at random, they prefer cats.}
\ptwo{A group of people were asked if they had run a red light in the last year. 150 responded ``yes'' and 185 responded ``no.'' Find the probability that if a person is chosen at random, they have run a red light in the last year.}
\ptwo{A U.S. roulette wheel has 38 pockets : 1 through 36, 0, and 00. 18 are black, 18 are red, and 2 are green. A play has a
dealer spin the wheel and a small ball in opposite directions. As the ball slows to stop, it can land with equal probability on
the 38 slots. Find the probability of the ball landing on green.}
\pone{A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is
chosen at random from the jar, what is the probability of choosing
\begin{enumerate}[(a)]
\item a red marble?
\item a green marble?
\item a blue marble?
\end{enumerate}}
\pone { Lisa has a large bag of coins. After counting the coins, she recorded the counts in the table below. She then decided to draw some coins at random, replacing each coin before the next draw.
\begin{center}
\begin{tabular}{ |c|c|c|c|}
\hline
Quarters & Nickels & Dimes & Pennies \\
\hline
27 & 18 & 34 & 21 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[(a)]
\item What is the probability that Lisa obtains a quarter on the first draw?
\item What is the probability that Lisa obtains a penny or a dime on the second draw?
\item What is the probability that Lisa obtains at most 10 cents worth of money on the third draw?
\item What is the probability that Lisa does not get a nickel on the fourth draw?
\item What is the probability that Lisa obtains at least 10 cents worth of money on the fifth draw?
\end{enumerate}}
\pone{Suppose you roll a pair of six-sided dice.
\begin{enumerate}[(a)]
\item List all possible outcomes of this experiment.
\item What is the probability that the sum of the numbers on your dice is exactly 6?
\item What is the probability that the sum of the numbers on your dice is at most 4?
\item What is the probability that the sum of the numbers on your dice is at least 9?
\end{enumerate}}
\ptwo{I asked my Facebook friends to complete a two-question survey. They answered the following questions:
Which beverage do you prefer in the morning: coffee or tea? What is your gender? I summarized the results in following table:
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& Coffee & Tea & \textbf{Total} \\
\hline
Female & 37 & 24 & \textbf{61} \\
\hline
Male & 22 & 31 & \textbf{53} \\
\hline
\textbf{Total } & \textbf{59 }& \textbf{55} & \textbf{114} \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[(a)]
\item What is the probability that I select a friend who prefers coffee?
\item What is the probability that I select a friend who is female?
\item What is the probability that I select a friend who is male and prefers tea?
\vspace{.8in}
\end{enumerate} }
\ptwo{A poll was taken of 14,056 working adults aged 40-70 to determine their level of
education. The participants were classified by sex and by level of education. The results
were as follows.
\begin{center}
\begin{tabular}{c|cc|c} \hline
Education Level & Male & Female & Total \\ \hline
High School or Less & 3141 &2434 & 5575 \\
Bachelor's Degree & 3619 &3761 & 7380 \\
Master's Degree & 534 &472 & 1006 \\
Ph.D. & 52 &43 & 95 \\ \hline
Total & 7346 &6710 & 14,056 \\
\end{tabular}
\end{center}
A person is selected at random. Compute the following probabilities:
\begin{enumerate}[(a)]
\item The probability that the selected person is a male
\item The probability that the selected person does not have a Ph.D.
\item The probability that the selected person has a Master's degree
\item The probability that the selected person is female and has a Master's degree
\end{enumerate} }
\end{exercises}
\end{comment}
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