4.137 Using probability rules.

Let , , and .

4.138 Work with a transformation.

Here is a probability distribution for a random variable :

4.139 A different transformation.

Refer to the previous exercise. Now let .

4.140 Roll a pair of dice two times.

Consider rolling a pair of fair dice two times. For a given roll, consider the total on the up-faces. For each of the following pairs of events, tell whether they are disjoint, independent, or neither.

4.141 Find the probabilities.

Refer to the previous exercise. Find the probabilities for each event.

4.142 Some probability distributions.

Here is a probability distribution for a random variable :

4.143 Wine tasters.

Two wine tasters rate each wine they taste on a scale of 1 to 5. From data on their ratings of a large number of wines, we obtain the following probabilities for both tasters' ratings of a randomly chosen wine:

4.144 Slot machines.

Slot machines are now video games, with winning determined by electronic random number generators. In the old days, slot machines were like this: you pull the lever to spin three wheels; each wheel has 20 symbols, all equally likely to show when the wheel stops spinning; the three wheels are independent of each other. Suppose that the middle wheel has eight bells among its 20 symbols, and the left and right wheels have one bell each.

4.145 Bachelor's degrees by gender.

Of the 2,325,000 bachelor's, master's, and doctoral degrees given by U.S. colleges and universities in a recent year, 69% were bachelor's degrees, 28% were master's degrees, and the rest were doctorates. Moreover, women earned 57% of the bachelor's degrees, 60% of the master's degrees, and 52% of the doctorates.³¹ You choose a degree at random and find that it was awarded to a woman. What is the probability that it is a bachelor's degree?

4.146 Higher education at two-year and four-year institutions.

The following table gives the counts of U.S. institutions of higher education classified as public or private and as two-year or four-year:³²

Convert the counts to probabilities, and summarize the relationship between these two variables using conditional probabilities.

4.147 Wine tasting.

In the setting of Exercise 4.143, Taster 1's rating for a wine is 3. What is the conditional probability that Taster 2's rating is higher than 3?

4.148 An interesting case of independence.

Independence of events is not always obvious. Toss two balanced coins independently. The four possible combinations of heads and tails in order each have probability 0.25. The events

may seem intuitively related. Show that so that and are, in fact, independent.

4.149 Find some conditional probabilities.

Choose a point at random in the square with sides and . This means that the probability that the point falls in any region within the square is the area of that region. Let be the coordinate and the coordinate of the point chosen. Find the conditional probability . (Hint: Sketch the square and the events .)

4.150 Sample surveys for sensitive issues.

It is difficult to conduct sample surveys on sensitive issues because many people will not answer questions if the answers might embarrass them. Randomized response is an effective way to guarantee anonymity while collecting information on topics such as student cheating or sexual behavior. Here is the idea. To ask a sample of students whether they have plagiarized a term paper while in college, have each student toss a coin in private. If the coin lands heads and they have not plagiarized, they are to answer No. Otherwise, they are to give Yes as their answer. Only the student knows whether the answer reflects the truth or just the coin toss, but the researchers can use a proper random sample with follow-up for nonresponse and other good sampling practices.

Suppose that, in fact, the probability is 0.3 that a randomly chosen student has plagiarized a paper. Draw a tree diagram in which the first stage is tossing the coin and the second is the truth about plagiarism.

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The outcome at the end of each branch is the answer given to the randomized-response question. What is the probability of a No answer in the randomized-response poll? If the probability of plagiarism were 0.2, what would be the probability of a No response on the poll? Now suppose that you get 39% No answers in a randomized-response poll of a large sample of students at your college. What do you estimate to be the percent of the population who have plagiarized a paper?

4.151 Blood bag demand.

CASE 4.2 Refer to the distribution of daily demand for blood bags in Case 4.2 (pages 210–211). Assume that demand is independent from day to day.

4.152 Risk pooling in a supply chain.

Example 4.39 (pages 232–233) compares a decentralized versus a centralized inventory system as it ultimately relates to the amount of safety stock (extra inventory over and above mean demand) held in the system. Suppose that the CEO of ElectroWorks requires a 99% customer service level. This means that the probability of satisfying customer demand during the lead time is 0.99. Assume that lead time demands for the Milwaukee warehouse, Chicago warehouse, and centralized warehouse are Normally distributed with the means and standard deviations found in the example.

4.153 Life insurance.

Assume that a 25-year-old man has these probabilities of dying during the next five years:

4.154 Risk for one versus many life insurance policies.

It would be quite risky for an insurance company to insure the life of only one 25-year-old man under the terms of Exercise 4.153. There is a high probability that person would live and the company would gain $875 in premiums. But if he were to die, the company would lose almost $100,000. We have seen that the risk of an investment is often measured by the standard deviation of the return on the investment. The more variable the return is (the larger σ is), the riskier the investment.