**by John Allen Paulos**

Over the years I’ve been teaching, many people have asked me about the content of an elementary course I teach. I’m interested in the syllabi and exams of courses in other fields, so this I hope may be of interest to others as well. The survey course on which this exam is based is a smorgasbord of probability, voting theory, scaling, and other variable material. Since the class is very large, I often reluctantly make the final exam multiple choice as is the example below. Try it if you like. Two hours is all the time you have. Writing useful prompts for ChatGPT will take too long to be of much help.

**Math Patterns Final Exam, Prof. Paulos**

1.) Two large screen TV’s are essentially the same except for their size. The width of the larger screen

is 48 inches and that of the smaller is 30 inches. The area of the larger screen is how many times the area of the smaller screen.

- 1.60 B 1.44 C 2.560 D 4.097

2.) The weight of the smaller TV is 44 pounds. What is the weight of the larger TV?

A 70.4 pounds B 112.64 pounds C 180.22 pounds

3.) Meteors that strike Earth always seem to land in craters, and fatal skiing accidents always seem to happen on the skier’s last run. Is the explanation for these:

A coincidence B reverse causation C Bayes Theorem D humorous flapdoodle

4.) A large pizza has a diameter 2.5 times the diameter of a small one of the same thickness. How many times as much pizza is in the larger one?

A 5 times as much B 6.25 times C15.625 times D 2.5 times

5.) If the national debt is about 30 trillion dollars, how much is each individual American’s share?

A $120,000 B $90,000 C $60,000

6.) In a certain company there are four stockholders, X, Y, Z, and W. X owns 48% of the stock, Y owns 25%, Z 24%, and W the remaining 3% of the stock. If 51% of the vote is needed to pass a measure, analyze the power situation among X, Y, Z, and W. In how many of the 16 situations (all the 2^{4} ways for the 4 stockholders to vote Yes or No) does W’s vote make a difference in the weighted outcome of the vote?

A 10 B 12 C 8 D 4 E 0

7.) The weights of 100 horse jockeys (people who race horses) are recorded and so are the weights of 100 middle school students. Which set of weights is likely to have greater variability.

A the jockeys B the middle school students C they’re equal

8.) The Earth is about 4.5 billion years old, and Homo Sapiens arouse approximately 200,000 years ago. If you shrink the 4.5 billion years to a single year, then when approximately did humans arrive on the scene?

A October 15 B early on December 10th C December 30 D late on December 31st

9.) The association of higher spelling test scores with elementary school students’ bigger shoe sizes is

A causal, but not correlational B correlational, but not causal C both D neither

10.) What kills more people each year in this country?

A homicides B car accidents C terrorism D plane crashes

11.) If 22% of the people in a certain neighborhood subscribe to the NY Times, 56 percent subscribe to the Philadelphia Inquirer, and 9 percent subscribe to both. What percent subscribe to *exactly *one of these papers?

A 78% B 69% C 60% D 63%

12.) There are two coins on the table, one fair, the other two-headed. You pick one coin at random, flip it twice, and note it comes up heads both times. Given this, what is the probability you chose the fair coin?

A 1/4 B. 1/3 C 1/5 D 2/7

13.) You can park in a lot every day for a certain fee, or you can risk parking in an illegal spot on the street. If you park illegally, you’ll get a $40 ticket about 20% of the time, about 2% of the time you’ll be towed and incur a $500 fine when you are, and the other 78% of the time, you’ll get away with it. On average, assuming these percentages remain constant, what will it cost you each day to park on the street?

A $18 B $40 C $28 D $16

14.) Flip a coin three times. You win $10 if heads come up once, $20 if heads come up twice, $30 if heads come up all three times, and you lose $200 if heads doesn’t come up at all. On average how much will you win or lose, each time you play this game?

A $120 B $32.50 C $110 D. **–**$10.

15.)Ten voters are trying to decide upon one of five candidates, A, B, C, D, or E. Their preference rankings are listed below.

**4 3 2 1 **

E D C B

B A A C

A B B D

D C D A

C E E E

Which of the five candidates is the Borda count winner?

A B C D E

16.) In general, is the ranked choice winner always the plurality winner?

A Yes B No

17.) Assume that most of the class does well on this test getting a variety of scores ranging from 75 to 100, but that 30% of the class get a zero on it. In this case which of the two numbers is likely to be the larger?

A the mean score B the median score

18.) In the pick 6 (out of 40 numbers) lottery, you pick the numbers 1,2,3,4,5,6, and your friend picks 3,8,9,24,31,39. Who is more likely to get the winning ticket?

A your friend B you C equally likely

19.) Roll a pair of dice. What is probability of getting a sum of exactly 5?

A 5/36 B 4/36 C 10/36 D 2/5

20.) What is the probability of getting a sum of exactly 5 given that the sum is at most 5?

A 5/36 B 4/36 C 10/36 D 2/5

21.) If a study indicates that 36% of ethnic group A and 45% of ethnic group B improves from some treatment, and a second study indicates that 60% of group A and 65% of group B improves, can one conclude that a higher percentage of group B improves from this treatment?

A Yes B No.

22.) What is more likely: getting four 6’s in arrow when rolling a single die or getting ten heads in a row when flipping a coin?

A equally likely B can’t say C ten heads D four 6’s

23.) If a roulette wheel (18 red, 18 black, 2 green sectors) is spun eight times, what is the probability of landing on red at least once?

A (18/38)8 B 8*(18/38) C 1-(20/38)8 D (20/38)8 E 8-(20/38)8

24.) Linda has a PhD in physics, was president of her senior class, and is an ardent traveler. Which of these is more likely?

A Linda works as a cashier at a convenience store

B Linda works as a cashier at a convenience store and is very active in a local women’s group

25.) Two people, George and Martha, predict the outcomes of 100 coin flips. George correctly predicts 52 of the 100 coin flips, and Martha correctly predicts 31 of the 100 coin flips. Whose performance, George’s or Martha’s, is most

impressive? That is, most in need of an explanation?

A George’s B Martha’s C same for both

26.) A couple plans on having 4 children. Which is more likely:

A They’ll have two boys and two girls B they’ll have 3 of one sex and one of the other.

27.) In a large hospital, 220 babies are born each month. Is it certain that every month there will be 2 or more babies having the same weight to within an ounce? (Assume newborn babies’ weights are less than 13lbs)

A Yes B No

28.) There are exactly three doors, A, B, and C, through which a dog can enter the kitchen. If the probability of entering through door A is 2 times the probability the dog will enter through door B, and the probability the dog will enter through B is 3 times the probability it will enter through door C, what is the probability it will enter through door B? (Let x equal the probability the dog enters through door C.)

A 2/9 B 1/3 C 3/10 D 4/9

29.) In a drawer there are 6 red, 6 blue, and 6 white socks. You reach in and pick socks at random. What is the smallest number you need to be certain of getting a red pair?

A 5 B 14 C 9 D18

30.) In a criminal trial what probability will the defense attorney stress

A the probability that the prints of an innocent person would match those at the scene of a crime

B the probability that a person whose prints match those at the scene would be innocent

C they’re the same

31.) House A is 6 blocks west and 7 blocks south of house B. In how many ways are there to get from A

to B (walking only along streets and no back-tracking or moving away)?

A 1,418 B 1,716 C 2,114

32.) Matilda has 8 friends and wants to invite 5 of them to a party. A complication is that two of them are feuding and so she can invites just one of them or neither of the, How many guest lists can she have?

A 46 B 22 C 36 D 56

***

*John Allen Paulos is a Professor of Mathematics at Temple University and the author of *Innumeracy* and *A Mathematician Reads the Newspaper. *His most recent book is *Who’s Counting –Uniting Numbers and Narratives with Stories from Pop Culture, Puzzles, Politics, and More*.*