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\textbf{\Large MAT 3701:  Assignment 3 \hfill My Name}
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\begin{enumerate}

\item Consider the following definitions:

Definition:  Two events $A$ and $B$ are \emph{independent} if $P(A \cap B) = P(A)P(B)$.  (That is, the probability that both $A$ and $B$ occur is the product of their probabilities.  Observe that this means the probability that $A$ occurs is unaffected by the knowledge that $B$ occurred.  Why?)

Definition:  Three or more events are \emph{pairwise independent} if each pair of them is independent.

Definition:  Three or more events are \emph{mutually independent} if the probability that all of them occur is the product of their probabilities.  (For example, events $A$, $B$, and $C$ are mutually independent if $P(A \cap B \cap C) = P(A)P(B)P(C)$.)

\begin{enumerate}

\item Suppose an urn contains four balls, labeled 1, 2, 3, and 4.   A ball is picked at random.   Thus, we can think of our sample space as the set $\{ 1, 2, 3, 4 \}$, where each element has probability $\frac{1}{4}$.  Let $A$ be the event that ball 1 or 2 is picked;  that is, $A = \{1, 2 \}$.  Let $B = \{1, 3 \}$, and let $C=\{1, 4 \}$.  Show that events $A$, $B$, and $C$ are pairwise independent but not mutually independent.

\item Can you create three events that are mutually independent but not pairwise independent?

\item Let $S$ be the square $\{(x,y): 0 \leq x \leq 1, 0 \leq y \leq 1\}$.  A point is picked at random from $S$.  Let $A$ be the event that $X \leq \frac{1}{2}$.  Let $B$ be the event that $y \leq \frac{1}{4}$.  Show that $A$ and $B$ are independent.

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\item If the half-life of a radioactive substance is 8 days and it is initially pure, what is the probability that an atom of the substance, selected at random, has decayed after 16 days?  After 4 days?  What is the probability it decays between day 4 and day 8?  In general, if $t$ is a positive real, what is the probability the atom survives for $t$ days?  (Recall that radioactive substances decay exponentially:  the rate of decay is proportional to the amount of substance remaining.)  What is the probability it has decayed after $t$ days?  What is the probability it decays between day $t_1$ and day $t_2$?

\item  Assume a light bulb is left on all the time and is equally likely to burn out at any moment.  (That is, the bulb does not ``age" or deteriorate;  the chance that it burns out during a given interval, if it was shining at the beginning of the interval, is proportional to the length of the interval.  This model is fairly realistic, at least for incandescent bulbs.)  Suppose the bulb has a 50\% chance of burning out in the first month after it is turned on.  What is the probability that the bulb will burn out in two months?  (Hint:  it can either burn out in the first month, or survive the first month but then burn out in the second.)  In half a month?

\item A box has 10 balls, 6 of which are black and 4 of which are white.  Assume that each ball is equally likely to be removed from the box.  Three balls are removed from the box, their color unnoted.  Find the probability that a fourth ball removed from the box is white.

\item Suppose there is a test for cancer with the property that 90\% of those with cancer react positively, whereas 5\% of those without cancer react positively. (The latter are \emph{false} positives.  Any test will produce some false positives.)  If a person reacts positively to the test, what is the probability that person has cancer?  What if the test is done twice and comes up positive both times?  If a person reacts negatively to the test, what is the probability that person has cancer that is not detected?  

\item Consider the cancer test above, and suppose that 1\% of patients in the population served by a clinic actually have cancer.  What is the probability that a random patient who tests positively at this clinic actually has cancer?  What if the test is done twice and comes up positive both times?  If a person reacts negatively to the test, what is the probability that person has cancer that is not detected?  What if the test is done twice, and the person reacts negatively at least once?  Negatively both times?

\item  Again considering the cancer test above (but a possibly different population than the one served by the clinic in the previous problem), suppose that 10\% of those tested randomly in a region react positively.  What is the cancer rate in the population of that region.

\item A certain component in a rocket engine fails 5\% of the time when the engine is fired.  To achieve greater reliability, this component is duplicated $n$ times and installed so that the engine fails only if all of these $n$ components fail.  Assume the component failures are independent of each other.  What is the smallest number $n$ that can be used to guarantee that the engine works 99\% of the time?

\item Suppose there are three chests each having two drawers.  The first chest has a gold coin in each drawer, the second chest has a gold coin in one drawer and a silver coin in the other drawer, and the third chest has a silver coin in each drawer.  A chest is chosen at random and a drawer opened.  If the drawer contains a gold coin, what is the probability that the other drawer also contains a gold coin?  (Be very,  very careful!)

\item A classic game show works has follows:  There are three curtains.  Behind one curtain is a fabulously valuable prize.  Behind the other two curtains are goats.  The contestant picks a curtain at random.  To create excitement, the host opens one of the other curtains, revealing a goat, and asks, ``Do you want to stick with your first choice, or switch to the remaining curtain?"  What is the best strategy for the contestant?  (Although my daughter would probably prefer the goat, this question assumes that the contestant is a crass materialist who would prefer the fabulously valuable prize.)

\item  Prove that if $A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots$ are events of a sample space with probability measure $\rho$, then $\rho(\cup_{n=1}^\infty A_n) = \lim_{n \rightarrow \infty} \rho(A_n)$.  (Hint:  Let $A^\prime_n = A_n \setminus A_{n-1}$, and consider $\cup_{n=1}^\infty A^\prime_n$.)  Explain why this limit always exists.

\item  Prove that if $A_1 \supseteq A_2 \supseteq A_3 \supseteq \cdots$ are events of a sample space with probability measure $\rho$, then $\rho(\cap_{n=1}^\infty A_n) = \lim_{n \rightarrow \infty} \rho(A_n)$.  Explain why this limit always exists.

\item Let $\alpha$ be a fixed irrational number.  Let the points of a circle be divided into equivalence classes as follows:  Two points are equivalent if one can be obtained from the other by rotating the circle through an angle of $n\alpha \pi$, where $n$ can be any integer.

Let $A$ be a set of points on the circle containing exactly one point from each equivalence class.  (\emph{Remark}:  Since there are infinitely many equivalence classes, choosing one element from each requires a powerful axiom from set theory called the \emph{Axiom of Choice}.)  A spinner is used to choose a point of the circle at random.  Is it possible to compute the probability that the spinner lands on a point in $A$? (In other words, is $A$ an event, meaning an element of the $\sigma$-algebra of sets that have probability measure in this sample space?) If so, what is it?

\emph{Hint:}  Let $\rho$ denote the probability measure on this the sample space $S$ of points chosen at random from the circle.  For $n \in \mathbb{N} \cup \{0\}$, let $A_n = \{x+n\alpha \pi | x \in A\}$;  that is, $A_n$ is the image of $A$ under rotation by the angle $n\alpha \pi$ (and, in particular, $A_0 = A$).  Clearly $\rho$ is invariant under rotation, since rotation does not change the portion of the circle occupied by any set of points, so $\rho(A_n) = \rho(A)$ for $n = 0, 1, 2, \ldots$.  Show that the sets $A_n$ are pairwise disjoint and that $\bigcup_{n=0}^\infty A_n= S$.  Argue from these claims that $\rho(A) \neq 0$, nor can we have $\rho(A) > 0 $ either;  hence $A$ is not measurable.

\end{enumerate}


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