Chapter 10 Exercises: Expected Value

Chapter: 10 — Expected Value: How Rational People Think About Risk Total Problems: 22 Difficulty Levels: 1 (Recall) through 5 (Synthesis)

Level 1: Recall and Recognition

These problems test your understanding of core definitions and concepts.

Problem 1.1 — Defining EV

In your own words (3–4 sentences), explain what expected value is and what it is NOT. Specifically: what does it tell you about a single decision, and what does it only tell you about a pattern of decisions?

Problem 1.2 — EV Calculation (Simple)

You flip a coin. If it lands heads, you win $8. If it lands tails, you lose $5. Calculate the expected value of this bet. Show your work using the formula EV = Σ(P(x) × V(x)).

Is this bet positive EV, negative EV, or zero EV? Should a rational player take it?

Problem 1.3 — Basic Vocabulary

Match each term to its correct definition:

Terms: a) Expected Value b) Variance c) Utility Function d) Kelly Criterion e) Outcome Bias

Definitions: 1. A formula for optimal bet sizing that balances growth against ruin risk 2. Evaluating a decision's quality based on its result rather than its reasoning 3. The probability-weighted average outcome across all scenarios 4. A curve mapping outcomes to their true subjective value 5. The spread of outcomes around the average; a measure of risk

Problem 1.4 — True or False

For each statement, indicate whether it is True or False and explain why in one sentence:

a) A positive EV bet should always be taken. b) Rational people sometimes buy insurance even though it has negative EV. c) The Kelly Criterion says you should always bet your entire bankroll on positive EV opportunities. d) Variance only matters if you plan to repeat a bet many times. e) The regret minimization framework and EV thinking are completely incompatible.

Problem 1.5 — St. Petersburg Paradox Preview

A game is played as follows: Flip a fair coin until it lands tails. Count the number of flips. You win $2^n where n is the number of flips.

a) What is the payout if tails appears on the first flip? b) What is the payout if you flip heads three times before getting tails? c) Why does this game have infinite expected value? d) Why do most people refuse to pay $1,000 to play this game?

Level 2: Application (Straightforward)

These problems apply EV concepts to concrete scenarios.

Problem 2.1 — Three-Outcome EV

A venture capital firm is considering an investment in a startup. They estimate: - 60% chance the startup fails completely, losing $1,000,000. - 30% chance the startup achieves modest success, returning $1,500,000. - 10% chance the startup is a major success, returning $8,000,000.

a) Calculate the expected value of this investment. b) Should the firm invest? What factors beyond pure EV would a rational investor consider? c) If the investment's cost is $1,000,000, what is the expected profit (EV minus cost)?

Problem 2.2 — Kelly Criterion Application

You have $500 in your poker bankroll. You've identified a situation where you have a 65% chance of winning a hand and the pot gives you exactly 1:1 odds (you win $1 for every $1 you bet).

a) Calculate the Kelly fraction (f* = (bp − q) / b) for this situation. b) How many dollars does the Kelly Criterion say you should bet? c) Why does Kelly not tell you to bet your entire $500, even though you have a clear edge? d) If you're uncertain whether your 65% estimate is accurate (maybe it's really 55% or 75%), how should that uncertainty affect your bet sizing?

Problem 2.3 — Job Decision EV

Nadia receives two job offers:

Offer A (Corporate Social Media Manager): - 90% probability: Stable position, $55,000/year, modest learning, low excitement - 10% probability: Excellent performance leads to promotion, $70,000/year within 2 years

Offer B (Startup Content Lead): - 45% probability: Startup thrives, $45,000/year growing to $65,000 + equity - 35% probability: Startup struggles but survives, $45,000/year for 18 months before finding new job - 20% probability: Startup fails, 6 months of job searching, zero income during search

a) Using Year 1 salary as the outcome value, calculate the EV of each offer. b) Now adjust your analysis: Nadia's parents have no financial safety net and she has $30,000 in student loans. How does this change your recommendation? c) Now consider: Nadia's primary career goal is to eventually build her own content brand. Which offer has higher non-monetary EV for this goal? Explain.

Problem 2.4 — Insurance EV

Marcus's chess tutoring app servers cost $200/month. He's considering buying server downtime insurance at $15/month. His research suggests there's a 5% monthly chance of a server outage that would cost him $350 in lost revenue and emergency tech support.

a) What is the expected monthly cost of server outages (without insurance)? b) What is the monthly cost of insurance? c) Compare these. Is buying insurance positive or negative EV? d) Despite the EV calculation, give two reasons Marcus might rationally buy the insurance anyway.

Problem 2.5 — Regret Minimization

Priya is offered a chance to spend four months working abroad on an unpaid creative project — a documentary about youth entrepreneurship. The opportunity costs are significant (no income, pause on job search). The potential benefits: strong portfolio piece, international experience, network expansion.

Apply Jeff Bezos's regret minimization framework: a) Imagine Priya at age 60 looking back. Write one paragraph describing how she might feel if she took the opportunity and it led nowhere professionally. b) Write one paragraph describing how she might feel if she declined and watched someone else do it. c) Which regret, in your assessment, is likely to be more painful? Does regret minimization give the same answer as EV analysis?

Level 3: Analysis

These problems require you to break down complex scenarios and identify underlying EV structures.

Problem 3.1 — Identifying Outcome Bias

Read the following three scenarios. For each, identify whether the evaluator is committing outcome bias and explain why:

a) Marcus runs a targeted ad campaign for his app. It fails to convert any users. He concludes his ad copy was poorly written and the strategy was a mistake.

b) Nadia posts a video on a new topic she's never tried before. It goes viral. She concludes she should always experiment with new topics.

c) Dr. Yuki folds a hand in poker with two pair — a normally strong holding — because she reads her opponent as having a set. Her opponent shows a set. She concludes she made the right decision.

Problem 3.2 — Building a Decision Matrix

Marcus is deciding whether to enter his chess tutoring app into a major startup competition. The entry requires 40 hours of preparation and a $500 entry fee.

Possible outcomes: - Win first place (5% probability): $10,000 prize, major press coverage worth ~$5,000 in equivalent marketing - Place in top five (15% probability): $2,000 prize, good press coverage worth ~$1,500 - Participate but don't place (80% probability): $0 prize, but valuable feedback from judges, networking with other founders (estimate value: $800)

a) Build a full decision matrix with all outcomes, probabilities, and values. b) Calculate the expected value of entering, accounting for the $500 entry fee as a cost. c) What is the EV of not entering (the counterfactual)? d) Make a recommendation. What factors beyond EV should Marcus consider?

Problem 3.3 — Kelly in Career Context

Apply the Kelly Criterion conceptually (not just mathematically) to the following career dilemma:

Priya has 100% of her job-search effort. She's debating whether to focus 90% on one "dream company" (Google) or spread her effort across 20 different companies. She estimates her probability of getting an offer from Google is roughly 3%.

a) Why might a 90% concentration violate the spirit of the Kelly Criterion? b) What does Kelly-style thinking suggest about how to distribute job application effort? c) Is there a career equivalent of "going bankrupt" in the job search context? What would it look like and how does it affect the optimal strategy?

Problem 3.4 — Utility Curve Analysis

Two people, both with $10,000 in savings, are offered the same bet: 50% chance to win $8,000, 50% chance to lose $6,000. EV = +$1,000.

Person A: Has a $30,000/year income, no debt, lives with parents. Person B: Has a $30,000/year income, $25,000 in credit card debt, rents an apartment.

a) For which person does a $6,000 loss represent a more severe utility hit? Explain using the concept of diminishing marginal utility. b) Should both people take this bet? Construct an argument for why their answers should differ even though the EV is identical for both. c) What does this tell us about the universality of "rational" decision advice?

Problem 3.5 — Multi-Round EV Dynamics

You start with $1,000. You can take the following bet repeatedly: 60% chance to win 50% of your current stake; 40% chance to lose 40% of your current stake.

a) Is this bet positive EV per round? (Calculate for a single round starting at $100.) b) What does the Kelly Criterion say about the optimal fraction to bet? c) If you bet your entire stake each round, what happens to your expected bankroll after many rounds? (Hint: think about what losing 40% many times does even if you win 60% of the time.) d) This is a famous result called the "Kelly-Breiman theorem." In your own words, explain the danger of maximizing EV per round without regard to bet sizing.

Level 4: Evaluation and Design

These problems require you to critique, evaluate, or design your own EV frameworks.

Problem 4.1 — Critiquing the Framework

A classmate says: "Expected value is just math. Real life is too unpredictable to use it. You'll never know the actual probabilities, so the calculation is meaningless."

Write a 300-word response that: a) Acknowledges the genuine limitation in their critique. b) Explains why EV thinking remains valuable even with imprecise probabilities. c) Gives a concrete example of a decision where rough EV thinking clearly outperforms pure intuition.

Problem 4.2 — Design Your Own EV Problem

Think of a real decision you are facing (or might face) in the next year — a choice about education, creative work, social risk, or career.

a) List all realistic outcomes (aim for 3–5). b) Assign your best probability estimate to each outcome. (They must sum to 1.0.) c) Assign a value to each outcome — you can use dollars, a utility scale of 1–10, or any other meaningful measure. Be honest. d) Calculate the expected value of both options. e) Does the EV calculation match your gut feeling? If not, why not?

Problem 4.3 — Evaluating Bezos's Regret Minimization

Jeff Bezos's regret minimization framework is presented in this chapter as an "alternative" to EV analysis. But is it really an alternative, or is it a special case of EV thinking?

Write a 400-word essay arguing one of the following positions: a) Regret minimization IS a form of EV thinking (with utility properly specified to include psychological future states). b) Regret minimization is genuinely different from EV thinking and captures something EV misses.

Use at least two specific examples to support your argument.

Problem 4.4 — The Variance-Survival Trade-off

You are the founder of a startup with 18 months of runway and $200,000 in the bank. You are presented with two strategic options:

Option A (Safe): Invest in standard customer acquisition (expected growth: 5%/month, very stable). Option B (Bold): Invest in a high-risk marketing campaign (30% chance of 40%/month growth, 70% chance of losing $80,000 with no growth).

a) Calculate the expected value of each option in terms of end-of-month cash position, starting from $200,000. b) Even if Option B has higher EV, argue why a founder might rationally choose Option A. c) Now flip the scenario: You have 3 months of runway and things are looking dire. Does your answer change? Explain the relationship between survival pressure and optimal risk-taking.

Level 5: Synthesis

These problems integrate EV concepts with the broader themes of the book.

Problem 5.1 — EV and the Science of Luck

The book argues that "luck is not a force — it's an outcome." How does expected value thinking reconcile this with the reality that lucky outcomes do happen?

Write a 500-word reflection on the following tension: "If I always make positive EV decisions, I will sometimes lose. Those losses will look like 'bad luck' to observers. Conversely, someone who makes systematically bad decisions might win through 'luck.' How should I think about luck in a world where EV governs decisions but outcomes are random?"

Your reflection should address: - The relationship between EV and luck - Whether good EV decisions deserve more credit than good outcomes - How this changes the way you evaluate your own and others' success

Problem 5.2 — Full Life Decision Matrix

Design a comprehensive EV framework for one of the following major life decisions:

a) Whether to pursue a traditional four-year degree vs. a vocational path vs. self-education b) Whether to move to a new city for a better career opportunity vs. stay near family and community c) Whether to start a small business vs. take a salaried job

Your framework should: 1. Identify at least 5 distinct outcome scenarios with probabilities and values 2. Include both monetary and non-monetary utility components 3. Apply the Kelly principle to resource allocation (time, money, energy) 4. Address the regret minimization question 5. Make a recommendation while acknowledging the limits of the analysis

Problem 5.3 — Poker Night Reflection (Integrative Essay)

Dr. Yuki says: "In poker, every decision — fold, call, raise — has an expected value. Professionals don't play hands. They play expected value across thousands of hands. A single session is noise. A career is signal."

Write a 600-word essay applying this quote to one of the following domains: a) Content creation (Nadia's world): How should a creator think about individual viral videos vs. the long-run EV of their content strategy? b) Startup building (Marcus's world): How should a founder think about individual product decisions vs. the long-run EV of their strategic positioning? c) Job searching (Priya's world): How should a job seeker think about individual application rejections vs. the long-run EV of their networking and positioning strategy?

Your essay should include at least one specific EV calculation, one application of the Kelly Criterion mindset, and one reference to the danger of outcome bias.

Selected answers to Problems 1.2, 2.1, 2.2, 2.4, and 3.2 appear in Appendix B.