CHAPTER 18SOLUTIONS AND MINI-PROJECT NOTES
CHAPTER 18
WHEN INTUITION DIFFERS FROM RELATIVE FREQUENCY
EXERCISE SOLUTIONS
18.1a.Both are equally likely, with probability (0.5)(0.5)(0.5)(0.5).
b.People would think the sequence MFFM had higher probability because it has 50% of each sex.
c.They are more likely to have two of each sex because there is only one sequence resulting in four girls (FFFF) but many resulting in two of each sex (MFFM, FMFM, etc.).
18.2See exercise 17 for an example. The main idea is that the probability of subsequent similar events actually does increase or decrease after observing an event.
18.3There are 50 49 pairs of students for which there could be a match and some names are extremely common.
18.4Any two people who compare a series of facts about themselves are likely to find some in common. Presumably the sisters would have compared notes on lots of factors. The names cited are heard quite frequently, so that adds to the likelihood of having them in common.
18.5As a rough calculation, there are over 290 million people in the United States, so even if the probability of dreaming of an airplane disaster is only 1 in a million, we would expect 290 people to have done so on any given night. However, if you were one of them and had never had such a dream before, you would probably think something amazing had occurred. The fact is that someone, somewhere will almost surely have such a dream before any crash.
18.6There are numerous meanings to the word "pattern" and it is easy to find something that fits at least one of them.
18.7a.They are saying that out of 11 who test positive only 1 is infected. Therefore the probability is 1/11 or about 9%.
b.You should convey the idea that there are a very large number of false positives because of the fact that most people who get tested do not have the disease. Even if there are no false negatives, the number of people who test positive because they have the disease would be low compared to the number who test positive because of a mistake in the test.
18.8In general businesses are more likely to base financial decisions on expected values, because their only concern is an overall profit, whereas individuals tend to incorporate emotional and psychological elements into their decisions. Also, businesses are more likely to have enough transactions to encounter the "long-run" average. (Either explanation is sufficient.)
18.9Yes, there are limited choices and people know their friends' calling behaviors.
18.10The best way to do this computation is to construct a table showing 100,000 hypothetical people in a population like the one described. Here is how it would look:
Test positive / Test negative / TotalActually sick / 95 / 5 / 100
Actually healthy / 9,990 / 89,910 / 99,900
Total / 10,085 / 89,915 / 100,000
Of the 10,085 testing positive, 95/10,085 = 0.0094 or 0.94% are sick.
18.11She is wrong and is suffering from the gambler's fallacy. Games like this have no memory, and they are not likely to reach the relative frequency of 40% in the short run.
18.12a.9,900/10,700 = 0.925
b.9,900/99,000 = 0.10
c.10,700/100,000 = 0.107
18.13Sensitivity is 800/1000 = 0.80. Specificity is 89,100/99,000 = 0.90. These correspond to the probabilities of correctly testing positive and correctly testing negative.
18.14There are thousands of hotels and motels with hundreds of rooms each. The probability that two people with the same name will sometimes occupy the same room on consecutive days must take into account millions of possibilities for that to occur. Adding the twist about asking for and receiving mail lowers the probability but would still not be too surprising.
18.15The probability of someone having a particular birthday, namely that of the professor, is much smaller than the probability that there will be some date in common for some pair in the class. He was probably not successful, because the probability for each student would be only 1/365. On average, we could expect 50/365 or 0.14 matches, which means that a match would occur in about 1 in 7 classes. (To be precise, the probability of no match is found by multiplying 364/365 fifty times, which is about 0.87.)
18.16The sensitivity of 0.90 is the probability of a positive test when someone has the disease, so the probability of a negative test when someone actually has the disease, i.e., a false negative, is 0.10. We cannot find the probability of a false positive.
18.17No. The probability of having to replace parts in a car in the near future is likely to depend on having to replace other parts. But it wouldn't be luck; it would be that she has already replaced the parts most likely to wear out first.
18.18There are numerous ways in which each statement could be true, and at least one of them would apply to almost everyone. For instance, the statement about the mother would apply if she was deceased, if she lived in a distant city, if there was friction in the relationship or if the person just wasn't as close to her as desired.
18.19Six heads in a row is just as likely as any of the other possibilities and there are numerous other sequences that might seem equally odd. So the total probability of an "unusual" sequence might be quite high. Specifying the sequence in advance means that the event of interest is only six heads in a row, which has probability (0.5)6 or 1/64 of happening.
18.20a.The expected value for choice A is $10 and for choice B it is $9, so people would choose A.
b.It is similar, because the expected value for buying a lottery ticket is less than the amount of money you keep by not buying the ticket, but by buying the ticket there is a chance of winning a large amount of money.
18.21They are offering equivalent deals in terms of the average cost of each suit. You may prefer the 50% off deal, because you have to buy only one suit to achieve the savings.
18.22a.Remember that if the odds are n to 1, then to break even the probability should be 1/(1+n). So in this case, the probability would have to be 1/3.
b.If the odds were 1 to 3 in favor of the other team you would have an expected value of zero.
c.If you bet $1 you could either gain $3 with probability 1/4, or lose $1.00 (i.e., "win" −$1) with probability 3/4. The expected value would be ($3)(0.25) + (−$1)(0.75) = 0.
18.23There are two possibilities. You pay $25 with probability 1/100 or pay nothing with probability 99/100, so the expected value is ($25)(0.01) = $0.25 or 25 cents. It will cost an average of 25 cents each time you risk getting a ticket, which seems like a cheap price for parking!
18.24a.The number 911 has the same probability as any other 3-digit number, which is 1/1000.
b.With 4 digits, the probability of any specific number is 1/10,000 because the choices are 0000 to 9999. However, any 4-digit number could have been repeated, so there are 1000 different ways to get the same number twice. Thus, the probability of getting the same number on consecutive drawings is 1000/10,000 = 1/1000. You can also see this by realizing that no matter what number was drawn first, the probability of getting that number on the next drawing is 1/1000.
NOTES ABOUT THE MINI-PROJECTS FOR CHAPTER 18
Mini-Project 18.1
Make sure a source is given for the information, because finding these numbers is the hardest part of this project. However, they are often available, and part of the purpose of the project is to make that fact known. One complication of this project is that sometimes the probabilities differ for different populations. If that's the case, the information should be conveyed in the project and one population can be chosen for the computations.
Mini-Project 18.2
The idea here is to try to approximate the probabilities using relative-frequency ideas and the rules from Chapter 15. Be sure the rules are properly applied. For instance, event probabilities should only be multiplied if the events can reasonably be assumed to be independent.
Mini-Project 18.3
Researchers have found that preferences are for a sure gain, but a gamble on the loss. Part a of this project simply asks for a comparison with those results. In part b it is important to explain how the survey was conducted in an unbiased way. The explanation should include the wording used for the introduction, to make sure participants were given a clue about the "desired" answer.
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