91Ó°ÊÓ

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?

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 ).

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.

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.)