Answer all 8 questions; you have exactly 75 minutes.

ALWAYS explain your reasoning

**1.** Answer this letter, assuming that each birth is an independent event
with an equal chance of being male or female: "If you have four children, they
may all be of one sex, there may be three of one sex and one of the other sex,
or there may be two of each. Which is most likely?" [letter to Marilyn vos Savant,
"Ask Marilyn," *Parade Magazine*, October 2, 1993.]

**2.** A researcher is going to roll m six-sided dice n times, each time recording the sum of the numbers on the m dice and using a histogram to display the results of these n experiments. Which of the following histograms is more likely to be bell-shaped: 200 rolls of 5 dice or 1000 rolls of 1 die? Explain.

**3.** For a woman who gives birth at age 35, the probability of having
a baby suffering from Down's syndrome is 1/270. A test of the amniotic fluid
in the mother's uterus is virtually 100 percent accurate in predicting Down's
syndrome, but costs about $1,000 and can cause a miscarriage. A study of the
effectiveness of an inexpensive blood test that does not risk miscarriage found
that in 89 percent of the Down's-syndrome cases, the test gave a positive reading,
while in 75 percent of the cases without Down's syndrome the test gave a negative
reading. Of those cases where there is a positive reading, what fraction are
false positives? [*The New England Journal of Medicine*, April 21, 1994.]

**4.** Explain why you agree or disagree with this claim by Edward Lazarus,
a partner in the Washington polling firm of Information Associates: "a randomly
selected sample would be in every way representative of the population at large
because every individual in the universe has an equal probability of being picked.
If that holds true, your attitudes, demographics, geography will be represented
proportionately in the sample." [Charles Kenney, *Boston Globe Magazine*,
August 30, 1987.]

**5.** Draw a bar graph and a histogram using these data on the number of previous prison sentences for men sent to prison in England and Wales in 1988. (Assume that the category 11 or more is 11-15) In what ways is the appearance of the bar graph misleading?

Previous sentences | 0 | 1-2 | 3-5 | 6-10 | 11 or more |

Number of persons | 2,562 | 2,808 | 4,911 | 7,977 | 10,716 |

**6.** On long automobile trips, Mrs. Jones drives and Mr. Jones gives directions. When there is a fork in the road, his directions are right 30 percent of the time and wrong 70 percent. Having been misled many times, Mrs. Jones follows Mr. Jones' directions 30 percent of the time and does the opposite 30 percent of the time. Assuming independence, how often do they drive down the correct fork in the road? If Mrs. Jones wants to maximize the probability of choosing the correct road, how often should she follow Mr. Jones' directions?

**7.** The annual returns on U.S. corporate stock and U.S. Treasury bonds over the next 12 months are uncertain. Suppose that these returns can be described by normal distributions with U.S. corporate stock having a mean of 15% and standard deviation of 20%, and U.S. Treasury bonds having a mean of 6% and standard deviation of 9%. If so, which asset is more likely to have a negative return? Explain your reasoning.

**8.** A certain college has found that half of the people they accept for admission decide to enroll and half do not. Assuming that these enrollment decisions are independent, with each student having a 1/2 probability of enrolling, how many students should they admit if they want the expected value of the number of students who enroll to be 400? If they follow this admission policy, what is the probability that more than 420 will enroll?