Practice Exercises: Chapter 8

Group 1: Probability Fallacies

Identify which fallacy or heuristic is at play in each scenario: Gambler’s Fallacy, Conjunction Fallacy, Availability Heuristic, or Base Rate Neglect.

1. A person refuses to swim in the ocean because they recently saw a movie about a shark attack, despite the statistical probability of a shark attack being 1 in 11.5 million.

2. “The roulette wheel has landed on red six times in a row. I’m putting all my money on black because it’s ‘due’ to hit!”

3. A doctor tells a patient they tested positive for a rare disease (1 in 10,000 people). The test is 99% accurate. The patient immediately assumes they are 99% likely to have the disease, ignoring how rare the disease is in the general population.

4. Participants are asked which is more likely: (A) That a massive earthquake will happen in North America, or (B) That a massive earthquake will happen in California causing a flood. Most people choose B.

Group 2: Evaluating Statistical Generalizations

Critique the following samples based on the three criteria: Size, Representativeness, and Randomness.

5. A student wants to know the campus opinion on tuition hikes. She interviews 200 students standing in line at the campus bookstore at 9:00 AM on a Monday.

6. To determine if a new local law is popular, a radio host asks listeners to “call in” and vote. 1,000 people call in, and 90% say they hate the law.

7. A scientist tests a new “memory pill” on 5 healthy male college students. All five show improved scores on a word-recall test. The scientist concludes the pill works for the general population.

Group 3: The “Average” Trap

Determine which measure of central tendency (Mean, Median, or Mode) would be the most honest or useful in the following situations.

8. You want to show the “typical” household income in a neighborhood where most people are middle-class but one resident is a multi-billionaire.

9. A clothing manufacturer needs to know which T-shirt size they should produce the most of for the upcoming season.

10. A teacher wants to see if the entire class is improving their math scores over the course of a year.

Group 4: Risk and Expected Value

Calculate the Expected Value (EV) and determine the “rational” choice based on the math.

11. You are offered a bet: A coin is flipped. If it’s heads, you win $100. If it’s tails, you lose $80. What is the EV? Should you take the bet?

12. A insurance policy costs $500 per year. It covers a specific risk that has a 1% (0.01) probability of occurring. If the event happens, the payout is $40,000. What is the EV of the policy? Is it “mathematically” worth the cost?

Answer Key

Group 1: Probability Fallacies

1. Availability Heuristic. The vivid memory of the movie makes the risk seem more probable than the statistics suggest.

2. Gambler’s Fallacy. The wheel has no memory; the probability remains 50/50 for red/black (ignoring the green zero).

3. Base Rate Neglect. The patient is ignoring the “Base Rate” (1 in 10,000). Mathematically, the chance they have it is actually quite low because the “false positive” rate of the test is likely higher than the frequency of the disease.

4. Conjunction Fallacy. (B) is a subset of (A). It is mathematically impossible for two conditions together to be more likely than one of those conditions alone.

Group 2: Statistical Generalizations

5. Weak Representativeness/Randomness. Students at a bookstore at 9 AM on Monday might not represent evening students, online students, or those who buy books elsewhere.

6. Weak Randomness (Selection Bias). This is a “Self-Selected” sample. People who call into radio shows usually have much stronger opinions than the average citizen.

7. Weak Size and Representativeness. 5 people is a “Hasty Generalization.” Furthermore, male college students do not represent the diverse ages and genders of the general population.

Group 3: The “Average” Trap

8. Median. The Median is robust against the billionaire “outlier,” providing a truer sense of the “middle” experience.

9. Mode. The manufacturer needs to know which specific size is requested most frequently.

10. Mean. The Mean is useful here because the teacher wants to see the aggregate movement of all scores combined.

Group 4: Risk and Expected Value

11. EV = $10. (0.50 x $100) + (0.50 x -$80) = $50 – $40 = +$10. Yes, the bet has a positive expected value.

12. EV = $400. (0.01 x $40,000) = $400. Since the policy costs $500 but only has a “value” of $400, it is not mathematically “worth it”—though a person might still buy it for peace of mind (Utility).