In this article
Before you start
This page sits in the statistics and probability learning path, next to the wider set of probability interview questions.
Why interviewers ask this
A question that begins "how much would you pay to play this game?" separates candidates who can turn a story into arithmetic from those who guess a price. The rest of the conversation then shows how far you can go beyond the arithmetic.
A weak answer names a number by feel. A strong answer lists the outcomes, multiplies each payoff by its probability, adds them, and explains that the result is a long-run average, not a promise about one play. A Staff-level answer also says when the rule strains: a game with an infinite expected value, a bank that cannot pay, or a player whose money has diminishing value to them. The interviewer's game can change from one interview to the next, so this page teaches the method and uses a die game and the St. Petersburg game as the two working cases.
Pricing a game from first principles
- Fair price
- A fair price is the entry fee at which a game breaks even for the player over many repeated plays, so that neither the player nor the house gains on average. It equals the game's expected value.
- St. Petersburg game
- The St. Petersburg game is a coin-flipping game in which a fair coin is flipped until the first head, the prize starts at 2 dollars and doubles with every tail before that head, so the expected prize is infinite although every actual prize is finite.
- Expected utility
- Expected utility is the probability-weighted average of how much a player values each outcome, rather than of the raw money, so that a gain matters less to a rich player than to a poor one.
In plain words
Imagine a stall that pays you whatever number a fair die shows. If you play for a very long afternoon, your average takings per roll settle near one figure. That figure is the most you can pay per roll and still not lose money. The analogy breaks in one place: it describes the long afternoon, not one roll. One roll pays 1 to 6, never the average.
In one precise sentence, the fair price of a game is the sum, over every outcome, of the payoff times the probability of that outcome.
Step 1: Expected value of a simple game gives the price
Expected value is the weighted average of a game's payoffs, with the probabilities as weights. Wikipedia's entry on expected value says that, provided it is finite, it can be read as the long-run average of independent repetitions. The law of large numbers is the result that the average of many independent repetitions settles near the expected value. The idea of a fair price equal to the expectation goes back to Huygens, who published his treatise in 1657, according to the same Wikipedia entry.
Take a die game: you roll once and receive the face value in dollars. The six faces are equally likely, each with probability 1/6, so the expected value is (1+2+3+4+5+6)/6 = 3.5.1 The fair price is 3.50 dollars. Pay less and you profit on average; pay more and you lose on average. On any single roll you still receive a whole number from 1 to 6. You now know the whole method: list outcomes, weight, add.
Step 2: The St. Petersburg game has an infinite expected value
In the St. Petersburg game, as the Stanford Encyclopedia describes it, a fair coin is flipped until it first lands heads. The prize is 2 dollars if that happens on flip one, 4 on flip two, 8 on flip three, and so on. The probabilities of those outcomes are 1/2, 1/4, 1/8 and so on.
Now apply Step 1. The first term is 1/2 × 2 = 1 dollar, the second 1/4 × 4 = 1 dollar, the third 1/8 × 8 = 1 dollar. Every term adds exactly one dollar, and there are infinitely many terms, so the sum has no limit.1 Wikipedia's article on the paradox draws the conclusion that, counting expected money alone, you should play at any price. Yet the Stanford entry stresses that the expected prize is not finite while the actual prize always is. The rule gave a silly answer; Step 3 shows why and what replaces it.
Step 3: A finite bank caps the price at about one dollar per flip
The first repair is that the bank is finite, a point Wikipedia's article on the paradox credits to Fontaine des Bertins in 1754. If the bank can pay for at most n flips, only n terms count and the expected value is n dollars. A bank of 2^30, about 1.07 billion dollars, covers 30 flips, so the game is worth 30 dollars.1 The same article's table gives about 20 dollars for a banker holding 1,050,000 dollars, 30 for a billionaire banker and 46 for world gross domestic product (GDP) in 2020. The table follows Wikipedia's convention: the game ends, and pays nothing more, when the bank can no longer cover the next flip.
| Flips the bank can cover | Bank needed (dollars) | Expected value (dollars) |
|---|---|---|
| 10 | 1,024 | 10 |
| 20 | 1,048,576 | 20 |
| 30 | 1,073,741,824 | 30 |
| 40 | 1,099,511,627,776 | 40 |
Text description of the diagram
- A bank that can cover 30 flips makes the game worth 30 dollars.
- Each extra flip the bank can cover adds exactly one dollar.
Step 4: Diminishing utility prices the game by wealth
The second repair is expected utility: money has diminishing value, so a player prices the average of how much each prize is worth to them. Gabriel Cramer (in a 1728 letter, using the square root of money) and Daniel Bernoulli (1738, using the logarithm of wealth) made this move, as the Stanford entry recounts; Bernoulli's own paper is Exposition of a New Theory on the Measurement of Risk. With log utility, our script, which solves the equation below, gives a maximum price of 20.87 dollars at a wealth of a million dollars, and a maximum price of 10.95 dollars at a wealth of 1,000 dollars.1 Wikipedia's paradox article reports 20.88 and 10.95.
Here w is your wealth and c the price; the largest c that keeps the left side at or above the right is your maximum.
Step-by-step description
- A die is worth 3.5.
- Each term adds one dollar.
- 30 flips are worth 30 dollars.
- Wealth sets the price.
- The fair die has an expected value of 3.5.
- Every St. Petersburg term adds one dollar, so the sum never ends.
- A bank that covers 30 flips makes the game worth 30 dollars.
- Log utility gives a maximum price of 10.95 dollars at a wealth of 1,000 dollars.
Knowing it is half the job. Saying it is the other half.
Interview Clozo is an AI interview copilot for real interview rounds: it hears the question and puts a strong answer on your screen, and you say it in your own words. See the features or how it works.
Start Free Trial →The Staff-level answer
How much should you be willing to pay to play this game?
I price it at its expected value: each payoff times its probability, summed. That is the break-even fee over many plays. For a die that pays its face value, it is 3.5. If the game is the St. Petersburg game, every term adds one dollar, so the sum is infinite, and no one should pay an unlimited fee. A real bank is finite, so a bank of about a billion dollars makes it worth about 30. And money has diminishing value, so with log utility someone holding a thousand dollars would pay about 11. Expected value is a long-run average, not what one play pays.
The numbers in that answer are the ones computed above.1 What makes it Staff-level is the last two moves: it says the textbook rule fails, names two repairs with their prices, and states what the expectation means.
The rubric
| Dimension | Weak answer | Strong answer | Staff answer |
|---|---|---|---|
| Correctness | Guesses a price | Computes the weighted sum | Computes it and checks it against a simple case |
| Depth | Gives only a number | Defines expected value as a long-run average | Adds the law of large numbers and the infinite case |
| Trade-offs | Ignores the bank and the player | Notes the bank is finite | Contrasts bank limits with diminishing utility |
| Communication | Jumps to the result | States outcomes, weights, sum | Says the structure aloud, then the caveat |
| Follow-up readiness | Surprised by "why not infinite?" | Knows the name of the paradox | Knows both repairs and what each leaves open |
Three variants of this question
What if you may re-roll the die once and keep the second roll?
The fair price is 4.25 dollars, against 3.5 for a single roll.1 Work backwards: price the last decision first, then step back to the earlier one. A re-roll is worth 3.5, so keep a first roll of 4, 5 or 6 and re-roll 1, 2 or 3. The value is (4+5+6)/6 + (1/2)(3.5) = 2.5 + 1.75 = 4.25.1 When a game's payoff depends on information revealed midway, condition on it first, as in the Bayes theorem questions; the same weighted-average idea prices an expected gain in A/B testing interview questions.
What if the bank only has a million dollars to pay out?
A bank of about 1.05 million dollars covers 20 flips, because 2^20 is 1,048,576, so the game is worth 20 dollars. With exactly one million dollars the bank covers 19 full flips (2^19 is 524,288), and paying out the whole bank after that adds about 1.9, so the value is about 20.9 dollars.1 This matches the 20 dollars Wikipedia's table gives for a banker holding 1,050,000 dollars.
What if you hold 1,000 dollars and value money by its logarithm?
Your maximum price is about 10.95 dollars, and a person holding 2 dollars would pay up to 3.35, borrowing if needed.1 The price rises with wealth because a given prize matters less to a richer player.
Common mistakes
- "Infinite expected value means I will win an infinite amount." (wrong) The actual prize is always finite. (right) "The expectation is infinite, but every single play pays a finite prize."
- "The expected value is what I get on one play." (wrong) A die never pays 3.5. (right) "It is the long-run average over many independent plays."
- "The paradox is a logical contradiction." (wrong) The Stanford entry on the paradox says that in a strict logical sense no formal contradiction is derived. (right) "It is a clash between a decision rule and intuition."
- "Log utility solves it completely." (wrong) The Stanford entry on the paradox notes that a version with prizes set in utility units brings the problem back. (right) "It helps, and it does not remove every version."
- "If people refuse the game, people are irrational." (wrong) The expected-utility entry separates what people do from what they should do. (right) "Prediction failing does not settle the normative question."
Follow-up questions
Why is the St. Petersburg game called a paradox?
Because an infinite expected value clashes with intuition: Wikipedia reports Ian Hacking as saying that few of us would pay even 25 dollars to enter. Nothing contradicts itself, so "paradox" is loose usage.
Does an expected value of 3.5 mean I win 3.5 on one play?
No. It is the long-run average over many repeats; a single roll pays a whole number from 1 to 6.
Is maximizing expected value the same as being rational?
Not by itself. Expected utility theory says to choose the act with the highest expected utility, and it makes faulty predictions about real choices; whether people should follow it is a separate question.
How would you check the answer with a simulation?
Simulate many plays and compare the average payout with the theory. Exactly half of all plays pay 2 dollars and three quarters pay at most 4, so any value from 2 to 4 is a median; in 100,000 simulated plays 87.5% paid at most 8, matching the exact 7/8.1 The sample mean wanders because the payout has no finite average, so a Monte Carlo estimate of it converges badly.
Can Chebyshev's inequality bound the payout?
No: Chebyshev's inequality needs a finite variance, and this game has no finite mean. Compare how the mean and median split here: the median is small while the mean is unbounded.
- The fair price of a game is its expected value: payoff times probability, summed.
- Expected value is a long-run average, not the result of one play.
- The St. Petersburg sum adds one dollar per term and never ends, though every prize is finite.
- A finite bank makes the game worth about one dollar per flip it can cover.
- Diminishing utility, such as logarithmic, prices the game by wealth; it does not fix every version.
Sources
- The St. Petersburg Paradox (Stanford Encyclopedia of Philosophy) — Stanford Encyclopedia of Philosophy (Martin Peterson), 1 Aug 2023
- Normative Theories of Rational Choice: Expected Utility — Stanford Encyclopedia of Philosophy, 26 Oct 2026
- St. Petersburg paradox — Wikipedia, 26 Oct 2026
- Expected value — Wikipedia, 26 Oct 2026
- Exposition of a New Theory on the Measurement of Risk (D. Bernoulli, 1738; translation in Econometrica 22(1):23-36, 1954) — Econometrica, 1 Jan 1954
