This is where you live or die on knowing the two-dice distribution cold.
For each proposition, calculate the exact probability using combinatorics, then compare it to the implied probability from the quoted odds. Bet only if your calculated probability exceeds the implied probability, indicating positive expected value. Explain your reasoning clearly and concisely, showing both the probability calculation and the EV comparison.
Pro tip: Always compute the implied probability from the odds (e.g., for odds of 5:1, implied probability is 1/(5+1) = 1/6) and compare it to your calculated probability. This directly shows whether the bet is favorable, and it's a quick way to impress interviewers with your practical understanding of odds.
Clarify the quoted odds for each proposition (e.g., '5 to 1' means you win $5 for a $1 stake, plus your stake back). Convert odds to implied probability using the formula: implied probability = 1 / (odds + 1) for 'odds against' format.
Use combinatorics to count favorable outcomes out of 6^5 = 7776 total outcomes. For example, sum equals 7: count the number of ways to get sum 7 with 5 dice (coefficient of x^7 in (x+x^2+...+x^6)^5).
For each proposition, if true probability > implied probability, the bet has positive expected value and you should bet; otherwise, do not bet. Compute expected value as (true probability * profit) - (1 - true probability) * stake, assuming a $1 stake.
State for each proposition whether to bet and why, summarizing the probability and EV comparison. Be prepared to explain any calculations succinctly.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First clarify the rules: two cards drawn without replacement from a standard 52-card deck, and the product of their values (Ace=1, Jack=11, Queen=12, King=13) must exceed 100. Then compute the exact probability by counting favorable ordered or unordered pairs, and compare the implied probability from 8:1 odds (1/9 ≈ 11.11%) to decide if the bet is favorable.
Pro tip: Show that you can quickly bound the probability: the maximum product is 13*12=156, and only high cards can contribute, so the probability is small. Then compute exactly to avoid guessing, and explicitly state the break-even probability for 8:1 odds.
Confirm that two cards are drawn without replacement, card values are Ace=1, Jack=11, Queen=12, King=13, and 'product exceeds 100' means strictly greater than 100. State whether order matters (it doesn't for the product).
List all unordered pairs of distinct card values (v1, v2) with v1*v2 > 100. For example, (9,12)=108, (10,11)=110, (10,12)=120, (11,10)=110, etc. Note that the minimum qualifying product is 108 (9*12) or 110 (10*11), so only high cards matter.
For each qualifying value pair, count the number of card combinations: if values are different, there are 4*4=16 ordered pairs (or 8 unordered pairs if counting unordered). Sum over all qualifying pairs to get total favorable outcomes.
Divide the number of favorable outcomes by the total number of possible outcomes (52*51=2652 ordered pairs, or 1326 unordered pairs). Simplify to a decimal or fraction.
Convert 8:1 odds to implied probability: 1/(8+1)=1/9≈11.11%. Compare the computed probability to this threshold. If the probability is greater than 11.11%, the bet is favorable; if less, it's unfavorable.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Kelly sizing is the obvious framework but the correlation across Game A bets completely breaks naive independent Kelly.
Frame the problem as a constrained portfolio optimization where all bets share a common risk factor (the roll), so you must account for correlation and maximize expected growth under a total budget. Walk through how you'd estimate edge and uncertainty for each proposition, then allocate using a risk-adjusted criterion like Kelly or mean-variance, explicitly handling the shared settlement. Close by discussing practical constraints like integer sizing, transaction costs, and robustness to model error.
Pro tip: Emphasize that because all bets settle on the same roll, diversification is limited—so you should size down relative to independent bets and consider hedging or prioritizing the highest-conviction propositions. Showing awareness of correlation's impact on optimal bet sizing signals real trading intuition.
Clarify that you want to maximize expected log wealth (or risk-adjusted return) subject to a total budget of 1,000 and the fact that all bets resolve on the same roll. Acknowledge that this is a single-period, correlated betting problem.
For each Game A proposition, estimate the probability of winning, the payout odds, and the uncertainty around your edge. This could come from historical data, a model, or market prices if available.
Because all bets settle on the same roll, specify the joint distribution of outcomes—e.g., which numbers on the die win for each proposition. This captures correlation and allows you to compute portfolio-level risk and return.
Use a portfolio optimization method (e.g., Kelly criterion for correlated bets, mean-variance, or simulation-based optimization) to find bet sizes that maximize the objective. Show that you'd shrink positions due to correlation and parameter uncertainty.
Round to feasible bet sizes, account for transaction costs or limits, and stress-test the allocation against model error or worst-case scenarios. Discuss how you'd monitor and adapt if new information arrives.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Expected value of two dice is 7, standard deviation around 2.4.
First, compute the expected value and variance of the sum of two fair dice, then set a bid-ask spread around that fair value based on your uncertainty and desired edge. When the interviewer keeps buying from you, interpret it as a signal that your ask is too low and adjust by widening the spread and skewing your quotes upward to protect against adverse selection.
Pro tip: In a two-sided market, the flow of trades is information. If you're consistently getting hit on the ask, it's not just bad luck—it's a signal that your price is wrong, so you should update your fair value and widen your spread to avoid being picked off.
Calculate the expected value of the sum of two fair dice: 7. Also note the distribution (e.g., variance) to understand the risk.
Choose a spread around the fair value that reflects your uncertainty and desired profit margin. For example, bid 6.5, ask 7.5.
If the interviewer keeps buying at your ask, treat it as a signal that your ask is too low. This is adverse selection: the buyer likely has better information or a different valuation.
Widen the spread and skew your quotes upward. For instance, raise both bid and ask, and increase the spread to protect yourself from further losses.
Articulate that you are managing inventory risk and information asymmetry. The adjustment is a standard market-making response to adverse flow.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Short answer: you can't just add up independent Kelly fractions.
Start by acknowledging that correlation means the bets are not independent, so the combined risk is higher than the sum of individual risks. Explain that you would size each bet smaller to account for the shared variance, and use a portfolio-level risk measure like Value at Risk (VaR) or expected shortfall to determine optimal sizes. Emphasize that the key is to avoid overbetting due to underestimating the joint probability of losses.
Pro tip: Mention that in practice, you'd use a copula or factor model to capture the dependence structure, and that the Kelly criterion for correlated bets involves the inverse of the covariance matrix. This shows you understand both theory and practical implementation.
Identify that all bets depend on the same dice roll, so their outcomes are perfectly correlated (or highly correlated if different bet types). Determine the joint distribution of outcomes.
Explain that correlation increases portfolio variance and the probability of simultaneous losses. The combined risk is not the sum of individual risks; it's higher due to positive correlation.
Use mean-variance optimization or Kelly criterion for correlated bets. The optimal bet size for each is smaller than if independent, and you may need to consider the bets as a single combined position.
Calculate portfolio VaR, expected shortfall, or other risk measures under the correlated scenario. Compare to the independent case to show the difference in required capital or bet size.
Discuss limits, margin requirements, and the possibility of hedging or diversifying across different dice rolls to reduce correlation risk.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Acknowledge that the optimal strategy depends on the payout structure and your utility function, not just expected value. Explain that when survival is at stake, maximizing expected value may be suboptimal if it increases the probability of ruin; instead, you might deliberately seek variance to maximize the chance of reaching a target, even at the cost of lower expected value. Tie this to decision-making under uncertainty and the importance of aligning strategy with objectives.
Pro tip: Optiver values pragmatic decision-making: show you understand that in trading, survival and capital preservation often trump raw EV, and that variance can be a tool when you're in a 'double-or-nothing' situation. Mention that you'd quantify the trade-off using expected utility or probability of reaching a goal.
Determine whether the goal is to maximize expected value, maximize probability of survival, or reach a specific target (e.g., double up). The answer hinges on what you're optimizing for.
Consider the game's rules: is it a fair bet, what are the odds, and what are the consequences of ruin? This shapes whether high-variance plays are beneficial.
Explain that with a concave utility function (risk-averse), you might avoid variance, but if you're in a 'must-double' situation, a convex utility (risk-seeking) can be rational to maximize chance of hitting the target.
Use probability of ruin or expected utility to compare strategies. For example, if you need to double up, a high-variance bet might give a 40% chance of success versus a 20% chance with a low-variance bet, even if EV is lower.
Make a recommendation based on the analysis, and explain that in practice, you'd adapt as new information arrives. Emphasize that deliberate variance-seeking can be optimal when the alternative is certain failure.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Overestimating edge inflates your Kelly fraction.
Start by defining the Kelly criterion and its assumptions, then explain why fractional Kelly is used in practice due to estimation error and risk aversion. Finally, detail the consequences of overestimating edge, emphasizing the rapid growth of overbetting risk and potential ruin.
Pro tip: Quantify the impact: overestimating edge by a factor of 2 can lead to betting twice the optimal fraction, which increases the probability of ruin and reduces long-term growth rate to zero or negative.
State that Kelly maximizes expected logarithmic utility of wealth, assuming known true probabilities and payoffs. It gives the optimal fraction to bet for long-term growth.
Discuss why traders bet a fraction (e.g., half-Kelly) due to parameter uncertainty, estimation error, risk of ruin, and non-log utility preferences. Fractional Kelly reduces volatility and drawdowns at a modest cost to growth.
If edge is overestimated, the Kelly fraction is too high. This leads to overbetting, which increases variance and can cause negative expected growth if the bet size exceeds twice the true Kelly fraction.
Use the formula for growth rate: g(f) = f * edge - f^2 * variance / 2. Overestimating edge leads to f > f*, reducing g(f) and potentially making it negative, leading to ruin.
Emphasize that fractional Kelly is a robust risk management tool that accounts for model uncertainty and avoids catastrophic losses from overestimation.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
It doesn't matter from an information standpoint since dice have no private signal.
First, clarify that in a dice game, the distinction between inventory management and information is blurred because the true value is known probabilistically and trades are anonymous. Then, explain that three consecutive lifts likely indicate information, but you must also consider inventory risk and adjust quotes accordingly. Finally, emphasize that the key is to update your beliefs and manage risk dynamically, regardless of the label.
Pro tip: In market-making interviews, always tie your answer back to risk management: show that you would widen spreads or skew quotes to protect against adverse selection, even if you suspect information.
State your assumptions about the dice game: e.g., is it a single die with known distribution, are trades anonymous, and is there a finite inventory? This sets the context for your reasoning.
Explain that inventory management is about controlling risk from holding a position, while information is about the counterparty knowing something you don't. In a dice game, information is probabilistic, so the line is fuzzy.
Consider that three consecutive lifts are unlikely under normal inventory-driven flow, suggesting informed trading. But also note that if your inventory is already skewed, you might be quoting aggressively, leading to lifts.
Describe how you would adjust: widen spreads, skew quotes away from the direction of the lifts, or reduce size. Emphasize that the response is similar whether it's information or inventory, because both require risk mitigation.
Argue that in a dice game, the distinction matters less because the underlying value is known probabilistically; what matters is updating your beliefs and managing risk. The label is less important than the action.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.