4.137 Using probability rules.

Let , , and .

- Explain why it is not possible that events and can be disjoint.
- What is the smallest possible value for ? What is the largest possible value for ? It might be helpful to draw a Venn diagram.
- If events and are independent, what is ?

4.138 Work with a transformation.

Here is a probability distribution for a random variable :

Value of | 1 | 2 |

Probability | 0.4 | 0.6 |

- Find the mean and the standard deviation of this distribution.
- Let . Use the rules for means and variances to find the mean and the standard deviation of the distribution of .
- For part (b), give the rules that you used to find your answer.

4.139 A different transformation.

Refer to the previous exercise. Now let .

- Find the distribution of .
- Find the mean and standard deviation for the distribution of .
- Explain why the rules that you used for part (b) of the previous exercise do not work for this transformation.

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.

- on the first roll, or more on the first roll.
- on the first roll, or more on the second roll.
- or less on the second roll, or less on the first roll.
- or less on the second roll, or less on the second roll.

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 :

Value of | 2 | 3 | 4 |

Probability | 0.2 | 0.4 | 0.4 |

- Find the mean and standard deviation for this distribution.
- Construct a different probability distribution with the same possible values, the same mean, and a larger standard deviation. Show your work and report the standard deviation of your new distribution.
- Construct a different probability distribution with the same possible values, the same mean, and a smaller standard deviation. Show your work and report the standard deviation of your new distribution.

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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:

Taster 2 | |||||
---|---|---|---|---|---|

Taster 1 | 1 | 2 | 3 | 4 | 5 |

1 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 |

2 | 0.02 | 0.07 | 0.06 | 0.02 | 0.01 |

3 | 0.01 | 0.05 | 0.25 | 0.05 | 0.01 |

4 | 0.00 | 0.02 | 0.05 | 0.20 | 0.02 |

5 | 0.00 | 0.01 | 0.01 | 0.02 | 0.06 |

- Why is this a legitimate assignment of probabilities to outcomes?
- What is the probability that the tasters agree when rating a wine?
- What is the probability that Taster 1 rates a wine higher than 3? What is the probability that Taster 2 rates a wine higher than 3?

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.

- You win the jackpot if all three wheels show bells. What is the probability of winning the jackpot?
- What is the probability that the wheels stop with exactly two bells showing?

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}

Public | Private | |
---|---|---|

Two-year | 1000 | 721 |

Four-year | 2774 | 672 |

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

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.

- What is the probability at least one bag will be demanded every day of a given month? Assume 30 days in the month.
- What is the interpretation of one minus the probability found part (a)?
- What is the probability that the bank will go a whole year (365 days) without experiencing a demand of 12 bags on a given 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.

- For a 99% service level, how much safety stock of the part SurgeArrester does the Milwaukee warehouse need to hold? Round your answer to the nearest integer.
- For a 99% service level, how much safety stock of the part SurgeArrester does the Chicago warehouse need to hold? Round your answer to the nearest integer.
- For a 99% service level, how much safety stock of the part SurgeArrester does the centralized warehouse need to hold? Round your answer to the nearest integer. How many more units of the part need to be held in the decentralized system than in the centralized system?

4.153 Life insurance.

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

Age at death |
25 | 26 | 27 | 28 | 29 |

Probability | 0.00039 | 0.00044 | 0.00051 | 0.00057 | 0.00060 |

- What is the probability that the man does not die in the next five years?
- An online insurance site offers a term insurance policy that will pay $100,000 if a 25-year-old man dies within the next five years. The cost is $175 per year. So the insurance company will take in $875 from this policy if the man does not die within five years. If he does die, the company must pay $100,000. Its loss depends on how many premiums the man paid, as follows:
Age at

death25 26 27 28 29 Loss $99,825 $99,650 $99,475 $99,300 $99,125 What is the insurance company's mean cash intake (income) from such polices?

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.

- Suppose only one person's life is insured. Compute standard deviation of the income that the insurer will receive. Find , using the distribution and mean you found in Exercise 4.153.
- Suppose that the insurance company insures two men. Define the total income as where is the income made from man . Find the mean and standard deviation of .
- You should have found that the standard deviation computed in part (b) is greater than that found in part (a). But this does not necessarily imply that insuring two people is riskier than insuring one person. What needs to be recognized is that the mean income has also gone up. So, to measure the riskiness of each scenario we need to scale the standard deviation values relative to the mean values. This is simply done by computing , which is called the
(CV). Compute the coefficients of variation for insuring one person and for insuring two people. What do the CV values suggest about the relative riskiness of the two scenarios?*coefficient of variation* - Compute the mean total income, standard deviation of total income, and the CV of total income when 30 people are insured.
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- Compute the mean total income, standard deviation of total income, and the CV of total income when 1000 people are insured.
- There is a remarkable result in probability theory that states that the sum of a large number of independent random variables follows approximately the Normal distribution even if the random variables themselves are not Normal. In most cases, 30 is sufficiently “large.” Given this fact, use the mean and standard deviation from part (d) to compute the probability that the insurance company will lose money from insuring 30 people—that is, compute . Compute now the probability of a loss to the company if 1000 people are insured. What did you learn from these probability computations?