Spot the Profitable Dice and Card Bets in a Market-Making Game

In a one-hour online round run as a market-making game, the interviewer presents a series of short games built on dice and playing cards. Each comes with stated odds or a price, and along the way you answer many small mental-arithmetic questions. Your job is to work out quickly which games are profitable for you and on which side to be. The report does not list the specific games, so this version asks you to demonstrate each skill the round exercises. ### Clarifying Questions - How are offers stated: odds against (such as "4 to 1"), a payout multiple that includes the stake, or a price for a contract that pays a fixed amount? - Are the dice fair and six-sided, and is a standard 52-card deck dealt without replacement? - Can I choose either side of a game, or only accept or decline? - Is there a limit on how much I can trade, and is the goal expected profit alone or profit with limited risk? ### Part 1 — Fair probabilities and values, fast For typical dice and card events (sums of dice, "at least one" events over several rolls, two- or three-card draws, games where you may choose to roll again), compute the probability or the expected payoff exactly, and fast enough to answer within seconds. ```hint Count the complement For an "at least one" event over several rolls or cards, count the ways it never happens. ``` #### What This Part Should Cover - Exact counting for dice sums and card draws, with and without replacement - Expected values, including linearity of expectation and games with a choice to continue - Mental-arithmetic habits that keep fractions exact until the end ### Part 2 — Is the offer profitable? Given the stated odds or price for a game, decide whether to take it, decline it or take the other side, and say how large your edge is per play. ```hint Turn every offer into a probability Before comparing anything, work out the win probability at which the offer would be exactly fair. ``` #### What This Part Should Cover - Converting each odds format to its break-even probability, and the expected value per unit staked - Choosing the side, and quantifying the edge - How variance and repetition affect how much you would play ### Part 3 — Make a market on a game Asked to quote a bid and an offer on the value of a game, set them around your fair value, and adjust them as dice are rolled or cards are revealed and as the counterparty trades with you. ```hint Who knows more? Ask whether the counterparty can know anything about the outcome that you do not, and what their trades should then change. ``` #### What This Part Should Cover - Quoting around fair value with a width that reflects uncertainty and risk - Updating fair value exactly when information is revealed - When the counterparty's trades should move your fair value, and when they should only shift your quotes because of your position ### What a Strong Answer Covers - Correct probabilities and expected values, stated as exact fractions and approximate decimals - Fluent conversion between odds formats and break-even probabilities - A clear decision, side and size for each game, with its edge - Quotes centred on fair value and updated consistently as information arrives - Speed with accuracy, and calm correction of mistakes ### Follow-up Questions - One game with a small positive edge may be played many times, another with the same edge only once. Does your decision or your size change? - Right after you buy, the counterparty offers to sell you more at the same price. What might that tell you? - Midway through, you realize your fair value for a game was wrong. What do you do with your quotes and your position?