Chapter 11 Exercises: The Birthday Problem and Other Probability Surprises

Chapter: 11 — The Birthday Problem and Other Probability Surprises Total Problems: 22 Difficulty Levels: 1 (Recall) through 5 (Synthesis)

Level 1: Recall and Recognition

These problems test your understanding of core definitions and results.

Problem 1.1 — The Core Numbers

Fill in the blanks from memory (or from the chapter's table):

a) In a group of __ people, there is approximately a 50% chance of a shared birthday. b) In a group of 57 people, the probability of a shared birthday is approximately _%. c) In a group of just 10 people, the probability of a shared birthday is approximately %. d) The key reason our intuition fails on this problem is that we naturally think about pairs involving (ourselves), not all ___ between all people.

Problem 1.2 — Monty Hall Basics

True or False — and explain why in one sentence for each:

a) In the Monty Hall problem, switching doors gives you a 50% chance of winning the car. b) After Monty opens a door, the car is equally likely to be behind either remaining door. c) The probability that your original door has the car never changes throughout the game. d) The Monty Hall problem would have a different answer if Monty randomly opened any door (including potentially the one with the car). e) The correct strategy is "always switch."

Problem 1.3 — Vocabulary Match

Match each term to its definition:

Terms: a) Inspection Paradox b) St. Petersburg Paradox c) Conditional Probability d) Collision Problem e) Friendship Paradox

Definitions: 1. A game with infinite expected dollar value but finite utility value, showing that dollar EV alone is insufficient for rational decisions 2. The general class of problems where many items drawn from a finite set produce matches much sooner than intuition expects 3. The mathematical result that most people have fewer friends than their friends have on average 4. The probability of event A given that event B has already occurred 5. The phenomenon where randomly-arriving observers experience longer-than-average intervals

Problem 1.4 — Six Degrees Math

If each person knows exactly 200 others, and there is no overlap in networks:

a) How many people are in your first-degree network? b) How many people are in your second-degree network? c) How many people are in your third-degree network? d) At what degree does the network theoretically exceed 8 billion (world's population)? e) Why does real-world overlap mean the actual degrees of separation are greater than this simplified math suggests?

Problem 1.5 — Coincidence Recognition

For each of the following "coincidences," identify which mathematical principle best explains why it was more probable than it seemed:

a) At a conference of 300 professionals in the same industry, two strangers discover they attended the same university at overlapping times. b) You've been waiting 20 minutes for a bus that runs every 10 minutes and feel the system is broken. c) You meet someone on a plane and discover you have a mutual Facebook friend. d) A game show contestant's initial door choice was wrong (as revealed at the end), making you wish they had switched when given the chance. e) A viral video creator's second video drastically underperforms their first major hit.

Level 2: Application (Straightforward)

These problems apply chapter concepts to concrete calculations.

Problem 2.1 — Classroom Birthday Problem

Your class has 30 students.

a) Using the complementary counting method, calculate the exact probability that at least two students share a birthday. (Show your setup even if you use a calculator.) b) If someone bet you $10 that no two students share a birthday, is this a bet you should take? Defend your answer. c) The teacher wants to run an experiment: she'll ask each student's birthday and check for matches. What's the probability she finds NO match? (Complement of your answer to (a).)

Formula reminder: P(no match among n people) = (365/365) × (364/365) × ... × ((365−n+1)/365)

Problem 2.2 — Monty Hall with Four Doors

Extend the Monty Hall problem: now there are four doors, one with a car and three with goats. You pick Door #1. Monty opens one goat door (Door #4). Now you can stay with Door #1 or switch to Door #2 or Door #3.

a) What is the probability the car is behind Door #1 after Monty's reveal? b) What is the combined probability the car is behind Door #2 or Door #3? c) If you switch (to either #2 or #3 randomly), what is your probability of winning? d) Compare the stay vs. switch winning probabilities. Is the advantage of switching larger or smaller (proportionally) than in the three-door version?

Problem 2.3 — The Generalized Birthday Problem

Suppose instead of birthdays (365 options), we're looking for two people who attended the same college from a pool of 1,000 colleges.

a) How many people do you need in a group before the probability of a shared-college pair exceeds 50%? (You may estimate or use the formula P ≈ 1 − e^(−n(n-1)/(2k)) where k is the number of categories and n is the group size, which approximates the exact calculation.) b) How does this compare to the birthday problem's answer of 23 people? c) In general: if we replace 365 "birthday" categories with k categories, how does the size of the group needed for a 50% collision probability change as k grows? (Qualitative answer is fine.)

Note: The 50% threshold is approximately n ≈ 1.18√k for any collision problem.

Problem 2.4 — Inspection Paradox in Practice

A campus shuttle runs at random intervals with a mean time of 12 minutes between shuttles (exponential distribution).

a) You arrive at a random time. What is the expected time until the next shuttle? (For an exponential distribution, the expected wait time for a random observer is equal to the mean interval — but explain why this feels longer than expected.) b) What is the expected size of the interval you landed in (the gap between the previous shuttle and the next one)? c) If 100 students are surveyed about how long they waited for the shuttle, would their reported average wait be longer or shorter than 6 minutes (half of 12)? Explain using the inspection paradox. d) A university administrator surveys students and concludes that the average shuttle wait is 14 minutes, not 6. She proposes increasing frequency. What should she actually compare this 14-minute figure against to determine whether the service is performing as designed?

Problem 2.5 — Boy-or-Girl Calculation

A family has two children. You learn that at least one is a girl.

a) List all possible two-child families (using B for boy and G for girl, ordered by birth). b) Which families are consistent with "at least one girl"? c) What is the probability both children are girls, given that at least one is a girl? d) Now the problem changes slightly: you meet a parent who says "My older child is a girl." Does the probability both are girls change? Calculate the new probability. e) Why do these two versions (at least one girl vs. the older child is a girl) give different answers?

Level 3: Analysis

These problems require deeper engagement with the mathematical structures.

Problem 3.1 — Coincidence Rate Analysis

Priya attends 5 networking events over two months. Each event has 150 attendees. She knows 300 people professionally. The city's tech sector has 40,000 professionals.

Using the approximation from the chapter's simulation framework:

a) Estimate the probability that at any given event, Priya finds at least one person she knows. b) Estimate the probability that at any given event, she finds at least one mutual connection (someone who knows someone she knows). c) After 5 events, what is the cumulative probability that she has found at least one "impossible coincidence"? d) Does your calculation make the coincidences feel more or less lucky? Explain.

Problem 3.2 — Why Monty Hall Feels Wrong

This problem asks you to identify and analyze the psychological mechanism behind Monty Hall resistance.

a) The most common wrong answer is "50/50 — it's one of two doors." Identify specifically what probability rule is being misapplied. b) Explain in your own words why "the host knows where the car is" changes the probability calculation relative to "the host randomly opens a door and happens to reveal a goat." c) In the random-host version: if the host randomly opens a door and reveals a goat (by coincidence), does switching now give you a 2/3 advantage? (Hint: calculate this carefully — the answer changes.) d) Write a one-paragraph explanation of the Monty Hall solution for a skeptical friend who insists it's 50/50. Your explanation must not use the words "probability" or "statistics."

Problem 3.3 — Birthday Problem in Social Media

Nadia has 3,200 followers on Instagram. Each follower has an average of 800 other accounts they follow.

a) If Nadia posts a video about a specific niche topic, and there are roughly 50,000 accounts on Instagram focusing on that niche, what is the approximate probability that two random followers of Nadia both follow the same other niche account? b) When Instagram's algorithm suggests Nadia to new users ("Suggested for You"), what birthday-problem-style calculation is the algorithm performing? c) Nadia notices that many of her followers seem to also follow the same 5–10 other accounts. Is this a surprising coincidence or a birthday problem outcome? Explain. d) How might Nadia use birthday-problem thinking to identify the best accounts to collaborate with?

Problem 3.4 — St. Petersburg Rationality

A venture capitalist offers you the following deal: you can invest in a startup at various amounts, with the following payout structure: - 50% chance the startup fails completely (lose your investment) - 25% chance the startup returns 2× your investment - 12.5% chance it returns 4× - 6.25% chance it returns 8× - And so on (each halving probability doubles the return multiple)

a) Show that this investment has infinite expected return (in dollar multiples). b) Despite infinite expected return, what finite price would you personally be willing to pay for a $1 slot in this investment? Explain your reasoning. c) Using the concept of diminishing marginal utility, argue why your answer in (b) is rational rather than irrational. d) How does this analysis apply to Marcus's decision about whether to invest his startup's entire development budget into a single high-risk, high-return product feature?

Problem 3.5 — Network Degrees Calculation

Marcus's chess tutoring app has 500 registered users. He estimates: - Each user knows ~50 other chess players they interact with online - The global online chess community has approximately 500,000 active players

a) How many degrees of separation does it take for Marcus's network (via his users) to theoretically reach the entire chess community? b) If Marcus connects with just the top 100 users by activity level (who each know ~200 other players), how does this change the calculation? c) Marcus wants to run a promotion where he reaches chess players who have never heard of his app. Why do birthday-problem/network dynamics suggest that targeting his existing users' connections (2nd-degree network) is more efficient than reaching the general chess public? d) Apply the friendship paradox: if Marcus surveys his users about how many chess connections they have, will the average be higher or lower than the true population average? Why?

Level 4: Evaluation and Design

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

Problem 4.1 — Designing a Coincidence Test

You want to test whether two people "coincidentally" meeting at a professional conference share a connection. Design a study that would tell you whether such connections are "surprising" (lower than expected by birthday problem math) or "unsurprising" (consistent with the birthday problem prediction).

Your design should specify: a) What data you would collect b) What your "baseline" expected probability would be c) How you would determine if observed coincidence rates are higher or lower than the birthday-problem baseline d) What a result showing "higher than expected" coincidences would actually mean

Problem 4.2 — The Anti-Intuition

All of the probability problems in this chapter violate common intuition. Write a 400-word essay arguing one of the following positions:

a) Our probability intuition evolved for a world radically different from the modern world, and the birthday-problem-style errors it causes are understandable but fixable through mathematical training.

b) The problem is not "wrong intuition" but "wrong reference class" — if we ask the right version of the question, our intuitions are correct. The errors come from asking questions ambiguously.

Use at least two specific examples from the chapter to support your argument.

Problem 4.3 — Evaluate the LinkedIn Algorithm

LinkedIn's "People You May Know" feature surfaces people you are likely to know based on shared connections, employers, and schools. From the perspective of this chapter:

a) Explain what birthday-problem calculation LinkedIn is performing. b) Is this feature increasing coincidences, or revealing existing coincidences? What is the difference, and does it matter for how users should interpret the suggestions? c) LinkedIn's algorithm sometimes surfaces people from your past who you haven't thought about in years. Is this "creepy" or mathematically expected? Argue from the birthday problem perspective. d) Design a thought experiment that would let you test whether LinkedIn's coincidences are genuinely unusual or simply the expected output of birthday-problem math.

Problem 4.4 — The Monty Hall Decision in Real Life

The Monty Hall problem is really about: you made a choice with limited information, new information became available, should you update?

Identify a real-world decision that has this structure. Specifically:

a) Describe the initial decision (analogous to choosing a door). b) Describe what new information becomes available (analogous to Monty opening a door). c) Explain why many people, in this real situation, resist updating their initial choice. d) Calculate (or estimate) whether updating (switching) is the rational choice in your example.

Suggestion: Consider applying this to job application strategy, content posting strategy, or startup pivoting.

Level 5: Synthesis

These problems integrate chapter concepts with the book's broader themes.

Problem 5.1 — The Coincidence Economy

Priya's networking evening, analyzed through the birthday problem, shows that "lucky" coincidences at events are mathematically expected — not surprising at all.

Write a 500-word analysis of the following claim: "Once you understand the birthday problem, networking events change from activities where you 'hope for lucky coincidences' to activities where you 'harvest mathematically inevitable connections.' This changes how you should approach them."

Your analysis should: 1. Explain what changes about the networking mindset 2. Identify at least two specific behaviors that should change (and why, using the math) 3. Address the counterargument: "Even if connections are mathematically expected, you still need luck to turn them into opportunities" 4. Connect this to Dr. Yuki's principle: "Luck is not a force — it's an outcome"

Problem 5.2 — Probability Surprise Design Challenge

Design your own "probability surprise" — a simple scenario where the correct probability is dramatically different from what most people would guess. Your problem must:

a) Be original (not from the chapter) b) Have a clearly counterintuitive correct answer c) Be solvable using basic probability techniques (complementary counting, conditional probability, or the birthday-problem structure) d) Have a clear "why does intuition fail here?" explanation

After designing the problem, solve it yourself and show your work.

Problem 5.3 — Integrative Essay: Probability Surprises and the Science of Luck

The chapter ends with this claim: probability surprises reveal that human intuition is "miscalibrated in systematic and predictable ways." Write a 600-word essay responding to the following:

"If our probability intuitions are systematically biased, does that mean 'lucky people' are just people whose biases happen to work in their favor? Or does understanding probability surprises like the birthday problem actually make you luckier — not just better at understanding luck retrospectively?"

Your essay should: - Reference at least three specific probability puzzles from the chapter - Connect to the concept of expected value from Chapter 10 - Address whether mathematical understanding or behavioral change is more important for luck improvement - Include a personal reflection on one situation in your own life that the birthday problem, Monty Hall, or inspection paradox could help explain

Selected answers to Problems 1.1, 2.1, 2.2, 2.5, and 3.4 appear in Appendix B.