conditional_probability.tex

\exercisesheader{}

% 13

\eoce{\qt{Joint and conditional probabilities\label{joint_cond}} P(A) = 0.3, 
P(B) = 0.7
\begin{parts}
\item Can you compute P(A and B) if you only know P(A) and P(B)?
\item Assuming that events A and B arise from independent random processes,
\begin{subparts}
\item what is P(A and B)?
\item what is P(A or B)?
\item what is P(A$|$B)?
\end{subparts}
\item If we are given that P(A and B) = 0.1, are the random variables giving rise 
to events A and B independent?
\item If we are given that P(A and B) = 0.1, what is P(A$|$B)?
\end{parts}
}{}

% 14

\eoce{\qt{PB \& J\label{pbj}} Suppose 80\% of people like peanut butter, 89\% 
like jelly, and 78\% like both. Given that a randomly sampled person likes peanut 
butter, what's the probability that he also likes jelly?
}{}

% 15

\eoce{\qt{Global warming\label{global_warming}} A Pew Research poll asked 
1,306 Americans ``From what you've read and heard, is there solid evidence that 
the average temperature on earth has been getting warmer over the past few 
decades, or not?". The table below shows the distribution of responses by party 
and ideology, where the counts have been replaced with relative frequencies.
\footfullcite{globalWarming}
\begin{center}
\begin{tabular}{ll  ccc c} 
                    &                           & \multicolumn{3}{c}{\textit{Response}} \\
\cline{3-5}
                    &                           & Earth is  & Not       & Don't Know    &   \\
                    &                           & warming   & warming   & Refuse        & Total\\
\cline{2-6}
                    & Conservative Republican   & 0.11      & 0.20      & 0.02      & 0.33  \\
\textit{Party and}  & Mod/Lib Republican        & 0.06      & 0.06      & 0.01      & 0.13 \\
\textit{Ideology}   & Mod/Cons Democrat         & 0.25      & 0.07      & 0.02      & 0.34 \\
                    & Liberal Democrat          & 0.18      & 0.01      & 0.01      & 0.20\\
\cline{2-6}
                    &Total                      & 0.60      & 0.34      & 0.06      & 1.00
\end{tabular}
\end{center}
\begin{parts}
\item Are believing that the earth is warming and being a liberal Democrat mutually 
exclusive?
\item What is the probability that a randomly chosen respondent believes the 
earth is warming or is a liberal Democrat?
\item What is the probability that a randomly chosen respondent believes the 
earth is warming given that he is a liberal Democrat?
\item What is the probability that a randomly chosen respondent believes the 
earth is warming given that he is a conservative Republican?
\item Does it appear that whether or not a respondent believes the earth is 
warming is independent of their party and ideology? Explain your reasoning.
\item What is the probability that a randomly chosen respondent is a 
moderate/liberal Republican given that he believes that the earth is not 
warming? 
\end{parts}
}{}

\D{\newpage}

% 16

\eoce{\qt{Health coverage, relative frequencies\label{health_coverage_rel_freqs}} 
The Behavioral Risk Factor Surveillance System (BRFSS) is an annual telephone 
survey designed to identify risk factors in the adult population and report 
emerging health trends. The following table displays the distribution of health 
status of respondents to this survey (excellent, very good, good, fair, poor) 
and whether or not they have health insurance.
\begin{center}
\begin{tabular}{rrrrrrrr}
& &  \multicolumn{5}{c}{\textit{Health Status}} &  \\ 
\cline{3-7}
                    &       & Excellent & Very good & Good      & Fair      & Poor      & Total \\ 
\cline{2-8}
\textit{Health}     & No    & 0.0230    & 0.0364    & 0.0427    & 0.0192    & 0.0050    & 0.1262 \\ 
\textit{Coverage}   & Yes   & 0.2099    & 0.3123    & 0.2410    & 0.0817    & 0.0289    & 0.8738 \\ 
\cline{2-8}
                    & Total & 0.2329    & 0.3486    & 0.2838    & 0.1009    & 0.0338    & 1.0000
\end{tabular}
\end{center}
\begin{parts}
\item Are being in excellent health and having health coverage mutually 
exclusive?
\item What is the probability that a randomly chosen individual has excellent 
health?
\item What is the probability that a randomly chosen individual has excellent 
health given that he has health coverage?
\item What is the probability that a randomly chosen individual has excellent 
health given that he doesn't have health coverage?
\item Do having excellent health and having health coverage appear to be 
independent?
\end{parts}
}{}

% 17

\eoce{\qt{Burger preferences\label{burger_preferences}} A 2010 SurveyUSA poll 
asked 500 Los Angeles residents, ``What is the best hamburger place in Southern 
California? Five Guys Burgers? In-N-Out Burger? Fat Burger? Tommy's Hamburgers? 
Umami Burger? Or somewhere else?'' The distribution of responses by gender is 
shown below. \footfullcite{burgers}
\begin{center}
\begin{tabular}{l p{4cm} r r r }
                    &                       & \multicolumn{2}{c}{\textit{Gender}} \\
\cline{3-4}
                    &                       & Male  & Female    & Total \\
\cline{2-5}
                    & Five Guys Burgers     & 5     & 6         & 11 \\
                    & In-N-Out Burger       & 162   & 181       & 343 \\
\textit{Best}       & Fat Burger            & 10    & 12        & 22 \\
\textit{hamburger}  & Tommy's Hamburgers    & 27    & 27        & 54 \\ 
\textit{place}      & Umami Burger          & 5     & 1         & 6 \\
                    & Other                 & 26    & 20        & 46 \\
                    & Not Sure              & 13    & 5         & 18 \\
\cline{2-5}  
                    & Total                 & 248   & 252       & 500
\end{tabular}
\end{center}
\begin{parts}
\item Are being female and liking Five Guys Burgers mutually exclusive?
\item What is the probability that a randomly chosen male likes In-N-Out the best?
\item What is the probability that a randomly chosen female likes In-N-Out the 
best?
\item What is the probability that a man and a woman who are dating both like 
In-N-Out the best? Note any assumption you make and evaluate whether you think 
that assumption is reasonable.
\item What is the probability that a randomly chosen person likes Umami best or 
that person is female?
\end{parts}
}{}

\D{\newpage}

% 18

\eoce{\qt{Assortative mating\label{assortative_mating}} Assortative mating is a 
nonrandom mating pattern where individuals with similar genotypes and/or 
phenotypes mate with one another more frequently than what would be expected 
under a random mating pattern. Researchers studying this topic collected data on 
eye colors of 204 Scandinavian men and their female partners. The table below 
summarizes the results.\footfullcite{Laeng:2007}
\begin{center}
\begin{tabular}{ll  ccc c} 
                                        &           & \multicolumn{3}{c}{\textit{Partner (female)}} \\
\cline{3-5}
                                        &           & Blue  & Brown     & Green     & Total \\
\cline{2-6}
                                        & Blue      & 78    & 23        & 13        & 114 \\
\multirow{2}{*}{\textit{Self (male)}}   & Brown     & 19    & 23        & 12        & 54 \\
                                        & Green     & 11    & 9         & 16        & 36 \\
\cline{2-6}  
                                        & Total     & 108   & 55        & 41        & 204
\end{tabular}
\end{center}
\begin{parts}
\item What is the probability that a randomly chosen male respondent or his 
partner has blue eyes?
\item What is the probability that a randomly chosen male respondent with blue 
eyes has a partner with blue eyes? 
\item What is the probability that a randomly chosen male respondent with brown 
eyes has a partner with blue eyes? What about the probability of a randomly 
chosen male respondent with green eyes having a partner with blue eyes?
\item Does it appear that the eye colors of male respondents and their partners 
are independent? Explain your reasoning.
\end{parts}
}{}

% 19

\eoce{\qt{Drawing box plots\label{tree_drawing_box_plots}} After an introductory 
statistics course, 80\% of students can successfully construct box plots. Of 
those who can construct box plots, 86\% passed, while only 65\% of those students 
who could not construct box plots passed.
\begin{parts}
\item Construct a tree diagram of this scenario.
\item Calculate the probability that a student is able to construct a box plot 
if it is known that he passed.
\end{parts}
}{}

% 20

\eoce{\qt{Predisposition for thrombosis\label{tree_thrombosis}} A genetic test is 
used to determine if people have a predisposition for \textit{thrombosis}, which 
is the formation of a blood clot inside a blood vessel that obstructs the flow of 
blood through the circulatory system. It is believed that 3\% of people actually 
have this predisposition. The genetic test is 99\% accurate if a person actually 
has the predisposition, meaning that the probability of a positive test result 
when a person actually has the predisposition is 0.99. The test is 98\% accurate 
if a person does not have the predisposition. What is the probability that a 
randomly selected person who tests positive for the predisposition by the test 
actually has the predisposition?
}{}

% 21

\eoce{\qt{It's never lupus\label{tree_lupus}} Lupus is a medical phenomenon where 
antibodies that are supposed to attack foreign cells to prevent infections 
instead see plasma proteins as foreign bodies, leading to a high risk of blood 
clotting. It is believed that 2\% of the population suffer from this disease. The 
test is 98\% accurate if a person actually has the disease. The test is 74\% 
accurate if a person does not have the disease. There is a line from the Fox 
television show \emph{House} that is often used after a patient tests positive 
for lupus: ``It's never lupus." Do you think there is truth to this statement? 
Use appropriate probabilities to support your answer.
}{}

% 22

\eoce{\qt{Exit poll\label{tree_exit_poll}} Edison Research gathered exit poll 
results from several sources for the Wisconsin recall election of Scott Walker. 
They found that 53\% of the respondents voted in favor of Scott Walker. 
Additionally, they estimated that of those who did vote in favor for Scott 
Walker, 37\% had a college degree, while 44\% of those who voted against Scott 
Walker had a college degree. Suppose we randomly sampled a person who 
participated in the exit poll and found that he had a college degree. What is the 
probability that he voted in favor of Scott Walker?
\footfullcite{data:scott}
}{}