Chapter 3

An investment analyst has tracked a certain bluechip stock for the past six months and found that on any given day, it either goes up a point or goes down a point. Furthermore, it went up on \(25 \%\) of the days and down on \(75 \%\). What is the probability that at the close of trading four days from now, the price of the stock will be the same as it is today? Assume that the daily fluctuations are independent events.

In a nuclear reactor, the fission process is controlled by inserting special rods into the radioactive core to absorb neutrons and slow down the nuclear chain reaction. When functioning properly, these rods serve as a firstline defense against a core meltdown. Suppose a reactor has ten control rods, each operating independently and each having an \(0.80\) probability of being properly inserted in the event of an "incident." Furthermore, suppose that a meltdown will be prevented if at least half the rods perform satisfactorily. What is the probability that, upon demand, the system will fail?

In 2009 a donor who insisted on anonymity gave seven-figure donations to twelve universities. A media report of this generous but somewhat mysterious act identified that all of the universities awarded had female presidents. It went on to say that with about \(23 \%\) of U.S. college presidents being women, the probability of a dozen randomly selected institutions having female presidents is about \(1 / 50,000,000\). Is this probability approximately correct?

An entrepreneur owns six corporations, each with more than \(\$ 10\) million in assets. The entrepreneur consults the U.S. Internal Revenue Data Book and discovers that the IRS audits \(15.3 \%\) of businesses of that size. What is the probability that two or more of these businesses will be audited?

The probability is \(0.10\) that ball bearings in a machine component will fail under certain adverse conditions of load and temperature. If a component containing eleven ball bearings must have at least eight of them functioning to operate under the adverse conditions, what is the probability that it will break down?

Suppose that since the early \(1950 \mathrm{~s}\) some ten-thousand independent UFO sightings have been reported to civil authorities If the probability that any sighting is genuine is on the order of one in one hundred thousand, what is the probability that at least one of the ten-thousand was genuine?

Doomsday Airlines ("Come Take the Flight of Your Life") has two dilapidated airplanes, one with two engines, and the other with four. Each plane will land safely only if at least half of its engines are working. Each engine on each aircraft operates independently and each has probability \(p=0.4\) of failing. Assuming you wish to maximize your survival probability, which plane should you fly on?

Two lighting systems are being proposed for an employee work area. One requires fifty bulbs, each having a probability of \(0.05\) of burning out within a month's time. The second has one hundred bulbs, each with a \(0.02\) burnout probability. Whichever system is installed will be inspected once a month for the purpose of replacing burned-out bulbs. Which system is likely to require less maintenance? Answer the question by comparing the probabilities that each will require at least one bulb to be replaced at the end of thirty days.

The great English diarist Samuel Pepys asked his friend Sir Isaac Newton the following question: Is it more likely to get at least one 6 when six dice are rolled, at least two 6 's when twelve dice are rolled, or at least three 6 's when eighteen dice are rolled? After considerable correspondence [see (167)], Newton convinced the skeptical Pepys that the first event is the most likely. Compute the three probabilities.

The gunner on a small assault boat fires six missiles at an attacking plane. Each has a \(20 \%\) chance of being on-target. If two or more of the shells find their mark, the plane will crash. At the same time, the pilot of the plane fires ten air-to-surface rockets, each of which has a \(0.05\) chance of critically disabling the boat. Would you rather be on the plane or the boat?

If a family has four children, is it more likely they will have two boys and two girls or three of one sex and one of the other? Assume that the probability of a child being a boy is \(\frac{1}{2}\) and that the births are independent events.

Experience has shown that only \(\frac{1}{3}\) of all patients having a certain disease will recover if given the standard treatment. A new drug is to be tested on a group of twelve volunteers. If the FDA requires that at least seven of these patients recover before it will license the new drug, what is the probability that the treatment will be discredited even if it has the potential to increase an individual's recovery rate to \(\frac{1}{2} ?\)

A computer has generated seven random numbers over the interval 0 to 1 . Is it more likely that (a) exactly three will be in the interval \(\frac{1}{2}\) to 1 or (b) fewer than three will be greater than \(\frac{3}{4}\) ?

Listed in the following table is the length distribution of World Series competition for the sixty-four series from 1950 to 2014 (there was no series in 1994 ). \begin{tabular}{cc} \hline \multicolumn{2}{c}{ World Series Lengths } \\ \hline Number of Games, \(k \quad\) Number of Years \\ \hline 4 & 13 \\ 5 & 11 \\ 6 & 14 \\ 7 & 26 \\ \hline Data fron: www.baseball-almanac.com \end{tabular} Assuming that each World Series game is an independent event and that the probability of either team's winning any particular contest is \(0.5\), find the probability of each series length. How well does the model fit the data? (Compute the "expected" frequencies, that is, multiply the probability of a given-length series times 64 ).

Suppose a series of \(n\) independent trials can end in one of three possible outcomes. Let \(k_{1}\) and \(k_{2}\) denote the number of trials that result in outcomes 1 and 2, respectively. Let \(p_{1}\) and \(p_{2}\) denote the probabilities associated with outcomes 1 and 2 . Generalize Theorem \(3.2 .1\) to deduce a formula for the probability of getting \(k_{1}\) and \(k_{2}\) occurrences of outcomes 1 and 2, respectively.

A corporate board contains twelve members. The board decides to create a five- person Committee to Hide Corporation Debt. Suppose four members of the board are accountants. What is the probability that the Committee will contain two accountants and three nonaccountants?

One of the popular tourist attractions in Alaska is watching black bears catch salmon swimming upstream to spawn. Not all "black" bears are black, though - some are tan-colored. Suppose that six black bears and three tancolored bears are working the rapids of a salmon stream. Over the course of an hour, six different bears are sighted. What is the probability that those six include at least twice as many black bears as tan-colored bears?

Country A inadvertently launches ten guided missiles-six armed with nuclear warheads-at Country B. In response, Country B fires seven antiballistic missiles, each of which will destroy exactly one of the incoming rockets. The antiballistic missiles have no way of detecting, though, which of the ten rockets are carrying nuclear warheads. What are the chances that Country B will be hit by at least one nuclear missile?

Each year a college awards five merit-based scholarships to members of the entering freshman class who have exceptional high school records. The initial pool of applicants for the upcoming academic year has been reduced to a "short list" of eight men and ten women, all of whom seem equally deserving. If the awards are made at random from among the eighteen finalists, what are the chances that both men and women will be represented?

A display case contains thirty-five gems, of which ten are real diamonds and twenty-five are fake diamonds. A burglar removes four gems at random, one at a time and without replacement. What is the probability that the last gem she steals is the second real diamond in the set of four?

Consider an urn with \(r\) red balls and \(w\) white balls, where \(r+w=N\). Draw \(n\) balls in order without replacement. Show that the probability of \(k\) red balls is hypergeometric.

Show directly that the set of probabilities associated with the hypergeometric distribution sum to 1 . (Hint: Expand the identity $$ (1+\mu)^{N}=(1+\mu)^{r}(1+\mu)^{N-r} $$ and equate coefficients.)

Show that the ratio of two successive hypergeometric probability terms satisfies the following equation, $$ \frac{\left(\begin{array}{c} r \\ k+1 \end{array}\right)\left(\begin{array}{c} w \\ n-k-1 \end{array}\right)}{\left(\begin{array}{l} N \\ n \end{array}\right)} \div \frac{\left(\begin{array}{l} r \\ k \end{array}\right)\left(\begin{array}{c} w \\ n-k \end{array}\right)}{\left(\begin{array}{c} N \\ n \end{array}\right)}=\frac{n-k}{k+1} \cdot \frac{r-k}{w-n+k+1} $$ for any \(k\) where both numerators are defined.

Urn I contains five red chips and four white chips; urn II contains four red and five white chips. Two chips are drawn simultaneously from urn I and placed into urn II. Then a single chip is drawn from urn II. What is the probability that the chip drawn from urn II is white? (Hint: Use Theorem 2.4.1.)

As the owner of a chain of sporting goods stores, you have just been offered a "deal" on a shipment of one hundred robot table tennis machines. The price is right, but the prospect of picking up the merchandise at midnight from an unmarked van parked on the side of the New Jersey Turnpike is a bit disconcerting. Being of low repute yourself, you do not consider the legality of the transaction to be an issue, but you do have concerns about being cheated. If too many of the machines are in poor working order, the offer ceases to be a bargain. Suppose you decide to close the deal only if a sample of ten machines contains no more than one defective. Construct the corresponding operating characteristic curve. For approximately what incoming quality will you accept a shipment \(50 \%\) of the time?

Suppose that \(r\) of \(N\) chips are red. Divide the chips into three groups of sizes \(n_{1}, n_{2}\), and \(n_{3}\), where \(n_{1}+\) \(n_{2}+n_{3}=N\). Generalize the hypergeometric distribution to find the probability that the first group contains \(r_{1}\) red chips, the second group \(r_{2}\) red chips, and the third group \(r_{3}\) red chips, where \(r_{1}+r_{2}+r_{3}=r\).

Some nomadic tribes, when faced with a lifethreatening contagious disease, try to improve their chances of survival by dispersing into smaller groups. Suppose a tribe of twenty-one people, of whom four are carriers of the disease, split into three groups of seven each. What is the probability that at least one group is free of the disease? (Hint: Find the probability of the complement.)

An urn contains five balls numbered 1 to 5 . Two balls are drawn simultaneously. (a) Let \(X\) be the larger of the two numbers drawn. Find \(p_{X}(k)\). (b) Let \(V\) be the sum of the two numbers drawn. Find \(p_{V}(k)\).

Suppose a fair die is tossed three times. Let \(X\) be the largest of the three faces that appear. Find \(p_{X}(k)\).

Suppose a fair die is tossed three times. Let \(X\) be the number of different faces that appear (so \(X=1,2\), or 3 ). Find \(p_{X}(k)\).

A fair coin is tossed three times. Let \(X\) be the number of heads in the tosses minus the number of tails. Find \(p_{X}(k)\).

Suppose a particle moves along the \(x\)-axis beginning at 0 . It moves one integer step to the left or right with equal probability. What is the pdf of its position after four steps?

Suppose that five people, including you and a friend, line up at random. Let the random variable \(X\) denote the number of people standing between you and your friend. What is \(p_{X}(k)\) ?

Urn I and urn II each have two red chips and two white chips. Two chips are drawn simultaneously from each urn. Let \(X_{1}\) be the number of red chips in the first sample and \(X_{2}\) the number of red chips in the second sample. Find the pdf of \(X_{1}+X_{2}\).

Suppose \(X\) is a binomial random variable with \(n=4\) and \(p=\frac{2}{3}\). What is the pdf of \(2 X+1\) ?

A fair die is rolled four times. Let the random variable \(X\) denote the number of 6 's that appear. Find and graph the cdf for \(X\).

A fair die is rolled four times. Let the random variable \(X\) denote the number of 6 's that appear. Find and graph the cdf for \(X\).

Suppose \(f_{Y}(y)=4 y^{3}, 0 \leq y \leq 1\). Find \(P(0 \leq\) \(\left.Y \leq \frac{1}{2}\right)\).

For the random variable \(Y\) with pdf \(f_{Y}(y)=\frac{2}{3}+\) \(\frac{2}{3} y, 0 \leq y \leq 1\), find \(P\left(\frac{3}{4} \leq Y \leq 1\right)\).

For persons infected with a certain form of malaria, the length of time spent in remission is described by the continuous pdf \(f_{Y}(y)=\frac{1}{9} y^{2}, 0 \leq y \leq 3\), where \(Y\) is measured in years. What is the probability that a malaria patient's remission lasts longer than one year?

Let \(n\) be a positive integer. Show that \(f_{Y}(y)=\) \((n+2)(n+1) y^{n}(1-y), 0 \leq y \leq 1\), is a pdf.

If \(Y\) is an exponential random variable, \(f_{Y}(y)=\) \(\lambda e^{-\lambda y}, y \geq 0\), find \(F_{Y}(y)\)

If the pdf for \(Y\) is $$ f_{Y}(y)= \begin{cases}0, & |y|>1 \\ 1-|y|, & |y| \leq 1\end{cases} $$ find and graph \(F_{Y}(y)\).

A continuous random variable \(Y\) has a cdf given by $$ F_{Y}(y)= \begin{cases}0 & y<0 \\ y^{2} & 0 \leq y<1 \\ 1 & y \geq 1\end{cases} $$ Find \(P\left(\frac{1}{2}

A random variable \(Y\) has cdf $$ F_{Y}(y)= \begin{cases}0 & y<1 \\ \ln y & 1 \leq y \leq e \\ 1 & e

The cdf for a random variable \(Y\) is defined by \(F_{Y}(y)=0\) for \(y<0 ; F_{Y}(y)=4 y^{3}-3 y^{4}\) for \(0 \leq y \leq 1 ;\) and \(F_{Y}(y)=1\) for \(y>1\). Find \(P\left(\frac{1}{4} \leq Y \leq \frac{3}{4}\right)\) by integrating \(f_{Y}(y)\).

Suppose \(F_{Y}(y)=\frac{1}{12}\left(y^{2}+y^{3}\right), 0 \leq y \leq 2\). Find \(f_{Y}(y)\).

In a certain country, the distribution of a family's disposable income, \(Y\), is described by the pdf \(f_{Y}(y)=\) \(y e^{-y}, y \geq 0\). Find \(F_{Y}(y)\).

The logistic curve \(F(y)=\frac{1}{1+e^{-y}},-\infty

Suppose that \(f_{Y}(y)\) is a continuous and sym. pdf, where symmetry is the property that \(f_{Y}(y)=f\) for all \(y\). Show that \(P(-a \leq Y \leq a)=2 F_{Y}(a)-1\).