Illustrative hypothetical examples from textbooks

• Purely mathematical -- X's and Y's without a story.
• Questions about iconic randomizers (coins, dice, roulette, draws from an urn) that one could actually do, but with no motivation to actually do in the real world.
• Word problems designed to illustrate mathematical calculation technique without paying attention to the appropriateness of the real-world model or the intrinsic interest of the real-world problem.
• Problems relating to real-world predictions one could actually check.

Our "local" textbook on introductory statistics (Freedman-Pisani-Purves-Adhikari Statistics) has an unusually large proportion of real data -- see this classification of its examples. I don't know any mainstream probability textbook with a nontrivial number of real-world examples.

11 hypothetical examples

1. A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability 5/8, French with probability 5/8, and art and French together with probability 1/4. What is the probability that he chooses mathematics? What is the probability that he chooses either art or French?
2. Take a stick of unit length and break it into three pieces, choosing the break points at random. (The break points are assumed to be chosen simultaneously.) What is the probability that the three pieces can be used to form a triangle?
3. There are n applicants for the director of computing. The applicants are interviewed independently by each member of the three-person search committee and ranked from 1 to n. A candidate will be hired if he or she is ranked first by at least two of the three interviewers. Find the probability that a candidate will be accepted if the members of the committee really have no ability at all to judge the candidates and just rank the candidates randomly. In particular, compare this probability for the case of three candidates and the case of ten candidates.
4. Each of the four engines on an airplane functions correctly on a given flight with probability .99, and the engines function independently of each other. Assume that the plane can make a safe landing if at least two of its engines are functioning correctly. What is the probability that the engines will allow for a safe landing?
5. A doctor assumes that a patient has one of three diseases d_1 , d_2 , or d_3 . Before any test, he assumes an equal probability for each disease. He carries out a test that will be positive with probability .8 if the patient has d_1 , .6 if he has disease d_2 , and .4 if he has disease d_3. Given that the outcome of the test was positive, what probabilities should the doctor now assign to the three possible diseases?
6. In London, half of the days have some rain. The weather forecaster is correct 2/3 of the time, i.e., the probability that it rains, given that she has predicted rain, and the probability that it does not rain, given that she has predicted that it won't rain, are both equal to 2/3. When rain is forecast, Mr. Pickwick takes his umbrella. When rain is not forecast, he takes it with probability 1/3. Find
   (a) the probability that Pickwick has no umbrella, given that it rains.
   (b) the probability that he brings his umbrella, given that it doesn't rain.
7. Luxco, a wholesale lightbulb manufacturer, has two factories. Factory A sells bulbs in lots that consists of 1000 regular and 2000 softglow bulbs each. Random sampling has shown that on the average there tend to be about 2 bad regular bulbs and 11 bad softglow bulbs per lot. At factory B the lot size is reversed -- there are 2000 regular and 1000 softglow per lot -- and there tend to be 5 bad regular and 6 bad softglow bulbs per lot. The manager of factory A asserts, "We're obviously the better producer; our bad bulb rates are .2 percent and .55 percent compared to B's .25 percent and .6 percent. We're better at both regular and softglow bulbs by half of a tenth of a percent each". "Au contraire", counters the manager of B, "each of our 3000 bulb lots con- tains only 11 bad bulbs, while A's 3000 bulb lots contain 13. So our .37 percent bad bulb rate beats their .43 percent". Who is right?
8. The Acme Super light bulb is known to have a useful life described by the density function f(t)=.01 e^{-.01t}, where time t is measured in hours.
   (a) Find the failure rate of this bulb (see Exercise 2.2.6).
   (b) Find the reliability of this bulb after 20 hours.
   (c) Given that it lasts 20 hours, find the probability that the bulb lasts another 20 hours.
   (d) Find the probability that the bulb burns out in the forty-first hour, given that it lasts 40 hours.
9. Suppose you toss a dart at a circular target of radius 10 inches. Given that the dart lands in the upper half of the target, find the probability that
   (a) it lands in the right half of the target.
   (b) its distance from the center is less than 5 inches.
   (c) its distance from the center is greater than 5 inches.
   (d) it lands within 5 inches of the point (0, 5).
10. An airline finds that 4 percent of the passengers that make reservations on a particular flight will not show up. Consequently, their policy is to sell 100 reserved seats on a plane that has only 98 seats. Find the probability that every person who shows up for the flight will find a seat available.
11. Suppose that the time (in hours) required to repair a car is an exponentially distributed random variable with [rate] parameter = 1/2. What is the probability that the repair time exceeds 4 hours? If it exceeds 4 hours what is the probability that it exceeds 8 hours?

Discussion

• biased coins do not exist.