Chapter 6 Exercises: Probability Intuition

Level 1: Recall and Comprehension

1.1 Define probability in both the frequentist and Bayesian interpretations. Give a real-world example where each interpretation is most natural to apply.

1.2 What is the gambler's fallacy? What is the hot hand fallacy? In what sense are they mirror images of each other?

1.3 List the five foundational probability principles introduced in this chapter. Give one real-world example of each.

1.4 What is base rate neglect? Give the medical test example in your own words, showing the calculation.

1.5 What is the "reference class problem"? Why does choosing the wrong reference class lead to wrong probability estimates?

Level 2: Application

2.1 A fair six-sided die has been rolled 10 times and has shown 1 every single time. What is the probability it shows 1 on the eleventh roll? Explain your answer and name the fallacy someone would commit by answering differently.

2.2 You're deciding whether to apply to a prestigious internship. You estimate you have a 3% chance of getting it, but the internship would be life-changing if you got it. The cost of applying is 4 hours of work. Calculate the expected value of applying if the internship would give you $50,000 worth of career benefit. Should you apply? What does this calculation not capture?

2.3 Nadia tracks her TikTok analytics for 4 weeks and finds that videos posted on Thursdays average 5,000 views, while all other days average 3,200 views. She concludes Thursday is her "lucky day." Using probability concepts from this chapter, evaluate her conclusion. What would you need to believe it?

2.4 You read that a new medical treatment has a "95% success rate" in a clinical trial. Using base rate reasoning, explain why this number alone is insufficient to evaluate the treatment. What additional information do you need?

2.5 Two events: A) Rain in London this December, B) Marcus winning his next chess tournament. Which pair of outcomes is more likely to be independent? Which is more likely to be dependent on other factors? Explain.

Level 3: Analysis

3.1 Analyze the multiple comparisons problem as it applies to social media analytics. If Nadia is tracking 7 days × 4 time slots × 3 content types = 84 possible combinations, and she checks all of them for correlation with views, how many spurious correlations would she expect to find just by chance at the 5% significance level? What are the implications for how she should interpret her findings?

3.2 Compare and contrast frequentist and Bayesian thinking as approaches to evaluating Priya's job search strategy. How would each framework approach the question "Is my application strategy working?"

3.3 The chapter claims people are "surprised by the wrong things." Analyze three recent news events you remember and categorize: were they surprising because of their base rate (genuinely rare) or because of availability heuristic (dramatic and memorable)? For each, estimate roughly how surprised the statistical base rate warrants.

3.4 "Calibrated probability means matching your stated confidence to your actual predictive accuracy." Design a method to test whether someone's probability estimates are well-calibrated. What would you measure, and how?

Level 4: Synthesis and Evaluation

4.1 You are advising Marcus on whether to make a significant pivot in his startup strategy. He has gathered 3 months of data showing that users who join via a specific referral channel are 40% more retained. Design the probability analysis he should run before acting on this finding. Consider: sample size, alternative hypotheses, base rates, and what level of evidence would justify changing strategy.

4.2 The chapter presents Bayesian updating as the correct way to change beliefs under uncertainty. But there's a challenge: your prior probability can have enormous influence on your posterior, especially with limited evidence. Design an argument for why priors are sometimes more problematic than helpful, and how to mitigate this.

4.3 Write a 400-word essay on the following: "Probability is not just a mathematical tool — it's a moral one. How we assign probabilities to outcomes affects how we treat people and distribute resources." Use examples from both individual decision-making and social policy.

Level 5: Research and Extension

5.1 Find a published study on probability calibration among different professional groups (doctors, meteorologists, economists, etc.). What does the research show about which groups are best and worst calibrated? What training or experience correlates with better calibration? Write a 600-word summary and analysis.

5.2 The "multiple comparisons problem" (also called the "problem of false discovery rate") is a significant issue in social science research. Research this topic and find at least one published example where multiple comparisons led to a false positive finding that was later corrected. Explain what happened, why it's hard to avoid, and what researchers do to mitigate it.

5.3 Research the base rate for the specific career path you're considering (or the most relevant one you can identify). What fraction of people who start on this path reach various milestones? What factors correlate with success? Use this data to build a preliminary Bayesian estimate of your own probability of success, and list what assumptions you're making.