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\title{Stochastic Processes}
\author{Summer Semester 2026, Exercise 2}
\date{Due Thursday, May 7, 2026}
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\begin{enumerate}


\item Recall the classical \emph{Monty Hall problem}:
\begin{quote}
Suppose you're on a game show, and you're given the choice of three
doors. Behind one door is a car, behind the others, goats. You pick a
door, say \#1, and the host, who knows what's behind the doors, opens
another door, say \#3, which has a goat. He says to you, ``Do you want
to pick door \#2?'' Is it to your advantage to switch your choice of
doors?
\end{quote}
Now suppose that it has been observed that Monty Hall, the host, has a
preference for door \#2: whenever it is possible to open door \#2
without showing the car, he will open this door.

Given this additional information, is it to your advantage to switch
if he opens door \#2 and you see a goat?  Is it to your advantage if
he opens door \#3 and you see a goat?  Perform a quantitative analysis
of this variation in terms of conditional probabilities.

\item (Humpherys \& Jarvis, Problem 5.7) Five of my friends come to
dinner and take their coats off at the door when they arrive, and my
ever-helpful son puts the coats away in his room.  The guests leave
one at a time, and when each one leaves, my son brings back a random
coat (selected uniformly) and gives it to them. Since the guests are
in a hurry, they each put on the coat given to them without noticing
whether it is correct, and then they leave.
\begin{enumerate}
\item What is the probability that the first guest gets the right
coat?
\item If the first to leave gets the right coat, what is the
probability that the second to leave will get the right coat?
\item What is the probability that every guest will get the right
coat?
\item If the first guest to leave gets the coat belonging to the
second guest, what is the probability that the second guest will get
the right coat?
\item If the first coat is wrong, but it is also not the second
guest's coat, what is the probability that the second guest will get
the right coat?
\item Without knowing the outcome of the first coat, what is the
probability that the second coat will be right?  Hint: consider using
the law of total probability.
\end{enumerate}

\item (Humpherys \& Jarvis, Problem 5.17) Let $X$ be the outcome of
the roll of a fair six-sided die.  Find the expectation and the
variance of $X$.

\item (Humpherys \& Jarvis, Problem 5.30) A biologist is collecting
kangaroo rats in the desert and she is hoping to find some with a
certain trait that occurs in 10\% of the general population of
kangaroo rats.  She collects 100 rats in total.  Assuming that each
sample is independent of the others:
\begin{enumerate}
\item Which distribution describes the exact number of rats that have
the trait?
\item What is the expected number of rats that will have the trait?
\item What is the probability that she will find exactly 30 rats with
the trait?
\end{enumerate}
Note: For (b) and (c), you may submit a Python program with output.
(Is it possible to get a quick pencil-and-paper answer?)

\end{enumerate}
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