Probability I: Introductory Ideas

Studies have shown that the chance of a new car being a "lemon"(One with multiple warranty problems) is greater for cars manufactured on Mondays and Fridays. Most consumers don't know on which day their car was manufactured. Assuming a 5-day production week, for a consumer taking a car at random from a dealer's lot,

Isaac T.Olduso, an engineer for Atlantic Aircraft, disagrees, with his supervisor about the likelihood of landing-gear failure on the company's new airliner.

Congressman Bob Forehead has been thinking about the upcoming midterm elections and has prepared the following list of possible developments in his career during the midterm elections:He wins his party’s nomination for reelection. He returns to his law practice. He is nominated for vice president. He loses his party’s nomination for reelection. He wins reelection. (a) Is each item on this list an “event” in the category of “Midterm Election Career Developments?” (b) Are all of the items qualifying as “events” in part (a) mutually exclusive? If not, are any mutually exclusive? (c) Are the events on the list collectively exhaustive?

Which of the following pairs of events are mutually exclusive? (a) A defense department contractor loses a major contract, and the same contractor increases its work force by 50 percent. (b) A man is older than his uncle, and he is younger than his cousins.

c) A baseball team loses its last game of the year, and it wins the World Series. (d) A bank manager discovers that a teller has been embezzling, and she promotes the same teller.

The scheduling offi cer for a local police department is trying to decide whether to schedule additional patrol units in each of two neighborhoods. She knows that on any given day during the past year, the probabilities of major crimes and minor crimes being committed in the northern neighborhood were 0.478 and 0.602, respectively, and that the corresponding probabilities in the southern neighborhood were 0.350 and 0.523. Assume that major and minor crimes occur independently of each other and likewise that crimes in the two neighborhoods are independent of each other. (a) What is the probability that no crime of either type is committed in the northern neighborhood on a given day? (b) What is the probability that a crime of either type is committed in the southern neighborhood on a given day? (c) What is the probability that no crime of either type is committed in either neighborhood on a given day?

The Environmental Protection Agency is trying to assess the pollution effect of a paper mill that is to be built near Spokane, Washington. In studies of six similar plants built during the last year, the EPA determined the following pollution factors:

The American Cancer Society is planning to mail out questionnaires concerning breast cancer. From past experience with questionnaires, the Cancer Society knows that only 15 percent of the people receiving questionnaires will respond. It also knows that 1.3 percent of the questionnaires mailed out will have a mistake in address and never be delivered, that 2.8 percent will be lost or destroyed by the post offi ce, that 19 percent will be mailed to people who have moved, and that only 48 percent of those who move leave a forwarding address. (a) Do the percentages in the problem represent classical, relative frequency, or subjective probability estimates? (b) Find the probability that the Cancer Society will get a reply from a given questionnaire.

McCormick and Tryon, Inc., is a “shark watcher,” hired by fi rms fearing takeover by larger companies. This fi rm has found that one of its clients, Pare and Oyd Co., is being considered for takeover by two fi rms. The fi rst, Engulf and Devour, considered 20 such companies last year and took over 7. The second, R. A. Venus Corp., considered 15 such companies last year and took over 6. What is the probability of Pare and Oyd’s being taken over this year, assuming that

The acquisition rates of both Engulf and Devour and R. A. Venus are the same this year as they were last year?
This year’s acquisition rates are independent of last year’s?

In each case, assume that only one fi rm may take over Pare and Oyd.

Solution

As the administrator of a hospital, Cindy Turner wants to know what the probability is that a person checking into the hospital will require X-ray treatment and will also have hospital insurance that will cover the X-ray treatment. She knows that during the past 5 years, 23 percent of the people entering the hospital required X-rays, and that during the same period, 72 percent of the people checking into the hospital had insurance that covered X-ray treatments. What is the correct probability? Do any additional assumptions need to be made?

An air traffi c controller at Dulles Airport must obey regulations that require her to divert one of two airplanes if the probability of the aircraft’s colliding exceeds 0.025. The controller has two inbound aircraft scheduled to arrive 10 minutes apart on the same runway. She knows that Flight 100, scheduled to arrive fi rst, has a history of being on time, 5 minutes late, and 10 minutes late 95, 3, and 2 percent of the time, respectively. Further, she knows that Flight 200, scheduled to arrive second, has a history of being on time, 5 minutes early, and 10 minutes early 97, 2, and 1 percent of the time, respectively. The fl ights’ timings are independent of each other. (a) Must the controller divert one of the planes, based on this information? (b) If she fi nds out that Flight 100 defi nitely will be 5 minutes late, must the controller divert one of the airplanes? (c) If the controller fi nds out that Flight 200 defi nitely will be 5 minutes early, must she divert one of the airplanes?

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Probability I: Introductory Ideas

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