You are in a live 45-minute trading-game round at a proprietary-trading / market-making firm. You start with a bankroll of **1,000** (game money). The game runs for **3–4 rounds**. In every round you are shown the same **three betting games**, each with propositions you can stake money on. The propositions stay roughly the same from round to round, but the **quoted odds change each round**. You have only **2–4 minutes per round** to decide how to allocate your money, and the interviewer questions every decision you make ("Why did you bet that much? Why did you skip that one?"). All payouts are quoted as net odds $b:1$ — a winning \$1 stake returns your stake plus \$$b$ of profit; a losing stake is forfeited entirely. This round's board: **Game A — two fair six-sided dice are rolled (all Game A propositions settle on the same single roll).** | Proposition | Payout | |---|---| | A1: the sum equals 7 | 4 : 1 | | A2: the sum is even | 1.1 : 1 | | A3: at least one die shows a 6 | 2.5 : 1 | | A4: the sum is 8 or more | 1.5 : 1 | | A5: the sum is 7 or less | 0.8 : 1 | **Game B — two cards are drawn without replacement from a shuffled standard 52-card deck.** Count A = 1, J = 11, Q = 12, K = 13, number cards at face value. | Proposition | Payout | |---|---| | B1: the product of the two card values is greater than 100 | 8 : 1 | **Game C — market making.** A contract settles at the sum of two fair dice. You must quote a two-sided market (a bid and an ask), and the interviewer may buy or sell contracts at your quotes before the dice are rolled. Work through the parts below the way you would in the live round — out loud, fast, and with numbers. ### Constraints & Assumptions - Bankroll starts at 1,000 and carries over between rounds; you may split it across any subset of propositions, including several propositions in the same game, or skip everything. - Dice are fair and the deck is fully shuffled; Game A, Game B, and Game C settle independently of each other, but all Game A propositions settle on one and the same roll. - You have 2–4 minutes per round; mental arithmetic is expected, and you may keep notes on paper (the propositions repeat across rounds — only the odds are re-quoted). - There is no requirement to bet; unstaked money is simply carried forward. ### Clarifying Questions to Ask - Are the payouts net odds (profit of $b$ per \$1 staked) or gross returns including the stake? Is a losing stake lost in full? - May I stake opposing propositions simultaneously (for example, both A4 and A5)? - What is the objective — maximize expected final wealth, maximize the probability of finishing above some threshold, or demonstrate decision process regardless of outcome? - Do the games settle after each round (so I know my bankroll before allocating the next round), and can I go bankrupt mid-game? - Is there a per-proposition stake limit, or can I put my entire bankroll on one line? ### Part 1 In under two minutes, go through propositions A1–A5 and decide for each whether you would put money on it. Justify every keep-or-skip decision with a number. ```hint Fair odds first For a payout of $b:1$ and win probability $p$, the EV of a \$1 stake is $p \cdot b - (1-p)$. Equivalently, a bet is +EV exactly when $b$ exceeds the fair odds $(1-p)/p$. Have the two-dice sum distribution ($1/36, 2/36, \dots, 6/36, \dots, 1/36$) memorized so each check takes seconds. ``` #### What This Part Should Cover - Correct win probabilities for all five propositions from the two-dice distribution. - The EV (or fair-odds comparison) and the resulting bet / no-bet call for each line, computed at conversational speed. - Discipline on the tempting long shot: a large payout is not an argument — only the sign of the EV is. - Noticing that A4 and A5 together cover every possible outcome of the roll. ### Part 2 You have about 60 seconds for Game B. Estimate the probability that the product of the two card values exceeds 100, and decide whether 8 : 1 is a price worth taking. An exact count is not expected live — show how you would prune and approximate under time pressure, then verify the exact number afterwards. ```hint Prune the deck What is the largest product a 7-or-lower card can participate in? That observation collapses the problem to a small subset of the deck. ``` ```hint Two-stage estimate Estimate the probability that both cards land in the surviving high ranks first, then estimate what fraction of those pairs actually clears 100. Multiplying the two rough numbers gets you within a percentage point of the truth. ``` #### What This Part Should Cover - A pruning argument that bounds which cards can matter before any counting starts. - A fast two-stage approximation with an explicit comparison against the breakeven probability implied by 8 : 1. - A clear final call (bet or pass) with an honest statement of the estimate's error bar. ### Part 3 Now allocate actual dollars for this round. How much of your 1,000 goes on each proposition you like — and is there anywhere on the board where you should be far more aggressive than a "normal" bet? ```hint Sizing formula For a single $b:1$ bet with win probability $p$, the Kelly fraction is $f^* = \frac{pb - (1-p)}{b}$ — edge over odds. Ask yourself what this formula does as the probability of losing approaches zero. ``` ```hint Check the pair Convert each payout to an implied probability $\frac{1}{1+b}$. What does it mean when the implied probabilities of two complementary propositions sum to less than 1? ``` #### What This Part Should Cover - Concrete Kelly (or fractional-Kelly) stake numbers for at least the risky +EV bets, with the reasoning for not going all-in on them. - Whether the candidate audits the board for any pair of complementary propositions whose implied probabilities fail to sum to 1, and — once such a structure is confirmed as riskless — reasons correctly about the stake ratio and the appropriate level of aggression. - Awareness that all Game A stakes are correlated (one roll decides them all), so per-bet Kelly fractions cannot simply be added. ### Part 4 Quote a bid and an ask for the Game C contract that settles at the sum of the two dice. The interviewer may hit your bid or lift your offer for several contracts before the roll. What do you quote, and how do you adjust if the interviewer keeps buying from you? ```hint Center, spread, then skew Center your quotes on the contract's theoretical value and set the spread from how much one settlement can hurt you — the sum's standard deviation is about 2.4. After each trade, think about inventory risk and adverse selection before re-quoting. ``` #### Clarifying Questions for this Part - What size may the counterparty trade at each quote, and may I attach a maximum size? - May I re-quote after each trade, or is one market binding for the whole round? - Does everything settle on a single roll, or do I carry inventory across multiple rolls? #### What This Part Should Cover - A defensible fair value and a spread justified by settlement variance, not an arbitrary number. - Correct P&L mechanics for being hit or lifted (what a short position loses when the roll comes in high). - Quote skewing and size limits in response to one-sided flow, distinguishing inventory management from information-based adjustment. ### What a Strong Answer Covers Across all parts, the interviewer is grading a repeatable decision process more than the final bankroll: - One consistent pipeline applied everywhere — probability, then EV / fair-odds comparison, then bet-or-pass, then size — narrated aloud at speed. - Risk discipline: never staking negative-EV lines no matter how attractive the payout, fractional sizing on risky edges, and maximum aggression reserved for locked-in profit. - Numeracy under time pressure: precomputed distributions, written notes reused across rounds, approximations with stated error bars. - Composure under interrogation: every stake carries a one-line quantitative justification when the interviewer pushes back. ### Follow-up Questions - All Game A propositions settle on the same roll. How does that correlation change your joint sizing versus treating each bet with an independent Kelly fraction? - Suppose after two rounds you are down to 400 and believe you must roughly double your bankroll to pass the round. Is maximizing EV still the right objective, or should you deliberately buy variance — and how? - Why do practitioners typically bet only a fraction (say half) of the Kelly stake? What specifically goes wrong when you overestimate your edge? - In Game C, the interviewer lifts your offer three times in a row. When is raising your quotes about inventory, and when is it about information — and does the distinction even exist in a dice game?