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\chead{\textbf{Homework 6}}
\lhead{Math 213, Spring 2026}
\rhead{Due Wednesday, 3/11 at 9:00am}

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\begin{document}

\noindent In problems $\S 7.1: 10-20$ (even), the sample space $S$ is the set of all possible five-card hands from a standard deck. By definition of an $r$-combination, we know that $|S| = {52 \choose 5}$.

\begin{problem}{\S 7.1 - 10}
What is the probability that a five-card poker hand contains the two of diamonds and the three of spades?
\end{problem}

\begin{problem}{\S 7.1 - 12}
What is the probability that a five-card poker hand contains exactly one ace?
\end{problem}

\begin{problem}{\S 7.1 - 14}
What is the probability that a five-card poker hand contains cards of five different kinds?
\end{problem}

\begin{problem}{\S 7.1 - 16}
What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
\end{problem}

\begin{problem}{\S 7.1 - 27(a)}
Find the probability of selecting exactly one of the correct six integers in a lottery, where the order in which these integers are selected does not matter, from the positive integers not exceeding 40.
\end{problem}

\begin{problem}{\S 7.1 - 36}
Which is more likely: rolling a total of 8 when two dice are rolled or rolling a total of 8 when three dice are rolled? Show your work.
\end{problem}

\begin{problem}{\S 7.2 - 8(a,c,d)}
What is the probability of these events when we randomly select a permutation of $\{ 1, 2, \dots, n \}$ where $n \geq 4$?
\begin{enumerate}
    \item[(a)] 1 precedes 2.
    \item[(c)] 1 immediately precedes 2.
    \item[(d)] $n$ precedes 1 and $n-1$ precedes 2.
\end{enumerate}
\end{problem}

\begin{problem}{\S 7.2 - 18}
\begin{enumerate}
    \item[(a)] What is the probability that two people chosen at random were born on the same day of the week?
    \item[(b)] What is the probability that in a group of $n$ people chosen at random, there are at least two born on the same day of the week?
    \item[(c)] How many people chosen at random are needed to make the probability greater than $1/2$ that there are at least two people born on the same day of the week?
\end{enumerate}
\end{problem}

\begin{problem}{\S 7.2 - 24}
What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up tails?
\end{problem}

\begin{problem}{\S 7.2 - 30}
Find the probability that a randomly generated bit string of length 10 does not contain a 0 if bits are independent and if
\begin{enumerate}
    \item[(a)] a 0 bit and a 1 bit are equally likely.
    \item[(b)] the probability that a bit is 1 is 0.6.
    \item[(c)] the probability that the $i$th bit is a 1 is $1/2^i$ for $i = 1, 2, 3, \dots, 10$.
\end{enumerate}
\end{problem}

\begin{problem}{\S 7.2 - 34}
Find each of the following probabilities when $n$ independent Bernoulli trials are carried out with probability of success $p$.
\begin{enumerate}
    \item[(a)] the probability of no successes.
    \item[(b)] the probability of at least one success.
    \item[(c)] the probability of at most one success.
    \item[(d)] the probability of at least two successes.
\end{enumerate}
\end{problem}

\begin{problem}{\S 7.2 - 36}
Use mathematical induction to prove that if $E_1, E_2, \dots, E_n$ is a sequence of $n$ pairwise disjoint events in a sample space $S$, where $n$ is a positive integer, then 
\[ p\left( \cup_{i=1}^n E_i \right) = \sum_{i=1}^n p(E_i).\]
\end{problem}

\begin{problem}{\S 7.3 - 2}
Suppose that $E$ and $F$ are events in a sample space and $p(E) = 2/3$, $p(F) = 3/4$, and $p(F~|~E) = 5/8$. Find $p(E~|~F)$.
\end{problem}

\begin{problem}{\S 7.3 - 8}
Suppose that one person in 10,000 people has a rare genetic disease. There is an excellent test for the disease; 99.9\% of people with the disease test positive and only 0.02\% without the disease test positive.
\begin{itemize}
    \item[(a)] What is the probability that someone who tests positive has the genetic disease?
    \item[(b)] What is the probability that someone who tests negative does not have the disease?
\end{itemize}
\end{problem}

\end{document}