This is the warm-up but don't underestimate it.
Frame the problem as expected value maximization using the Kelly criterion, then discuss practical constraints like risk of ruin and time limits. Emphasize precomputing a lookup table of optimal bet fractions for all possible quoted odds to enable rapid decisions.
Pro tip: Mention that in a timed setting, you might use a fractional Kelly (e.g., half-Kelly) to reduce variance and avoid ruin, and precompute thresholds for when to bet at all.
Clarify that the coin is fair (50% win probability) and that quoted odds are given as decimal or fractional odds. Assume you can bet any fraction of your bankroll.
For given odds, calculate the expected profit per unit bet. Determine if there is a positive edge (EV > 0) to decide whether to bet.
Use the Kelly formula to find the fraction of bankroll that maximizes long-term growth. For a fair coin, the formula simplifies to f = 2p - 1, where p is the probability of winning (0.5), but adjust for odds.
Create a table mapping quoted odds to optimal bet fractions (and whether to bet). This allows O(1) decisions during the game.
Discuss using fractional Kelly to manage risk, setting minimum edge thresholds, and handling discrete bet sizes or limits.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
The triangular distribution with mean 7 is the thing to know cold.
Start by defining the sample space for two fair dice (36 equally likely outcomes) and the distribution of the sum, which is triangular. Explain that the probability mass function can be computed in O(1) time using a closed-form formula, making pricing fast. Then discuss how to price bets by calculating expected payouts or fair odds based on these probabilities.
Pro tip: Mention that the sum distribution is symmetric and can be generated via convolution, but the closed-form formula avoids simulation and is key for low-latency pricing in trading. Also, relate it to real-world betting: fair odds are the inverse of probabilities, and any house edge is applied on top.
State that each die is fair with outcomes 1-6, so there are 36 equally likely ordered pairs. The sum ranges from 2 to 12, and the number of ways to get sum s is given by a triangular function: 6 - |s - 7| for s in 2..12.
Compute probabilities as counts/36. For example, P(sum=7)=6/36=1/6, P(sum=2)=1/36. Emphasize that this is a closed-form O(1) computation, not requiring enumeration.
For a bet paying out based on the sum, calculate the expected payout by summing over outcomes: E = Σ payout(s) * P(s). Fair price is the expected payout (or fair odds = 1/P(event) - 1 for binary bets).
The sum of two dice follows a discrete triangular distribution, which is the convolution of two uniform distributions. This fact allows fast computation of probabilities for any event defined on the sum without simulation.
Mention how to handle more dice (e.g., via generating functions or normal approximation) and the importance of speed in pricing for market making. Also note that for events not solely dependent on sum, the joint distribution may be needed.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
E[XY] = E[X] times E[Y] for independent draws, and mean card rank is 7 so the product expectation is 49.
Start by defining the random variables for the card values and the product. Explain that the expected value of the product can be computed using the linearity of expectation if the draws are independent, or more generally by leveraging the property of conditional expectation. Emphasize that independence (or conditional independence) allows you to avoid enumerating all outcomes by factoring the expectation.
Pro tip: In trading interviews, always clarify whether draws are with or without replacement, as this drastically changes the calculation. Mentioning the difference shows attention to detail and practical trading intuition.
Let X and Y be the values of the two drawn cards. Clearly state whether the draws are independent (with replacement) or dependent (without replacement).
We want E[XY], the expected value of the product of the two card values.
If X and Y are independent, E[XY] = E[X] * E[Y]. If not, use E[XY] = E[E[XY|X]] = E[X * E[Y|X]] to reduce the problem to a single sum over X.
Calculate E[X] and E[Y] (or E[Y|X]) using the distribution of card values. For a standard deck, the average card value is 7.
Multiply the expectations (or average over X) to get the final expected product, and explain why this avoids enumerating all 52*51 outcomes.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Mean rank is 7, so fair value starts at 21.
Start by computing the expected value of a single card from a standard deck (7.0) and multiply by 3 to get the initial fair value (21). Then explain how to update the estimate as each card is revealed: subtract the revealed card's value from the remaining sum and divide by the number of remaining cards to get the new expected value for the next card, then add to the sum of already revealed cards. Emphasize that this is a conditional expectation problem and that the fair value is the expected sum given all available information.
Pro tip: Mention that the order of revelation doesn't affect the final expected sum, but the conditional expectation updates are crucial for pricing and risk management. Also, note that if the deck composition changes (e.g., cards removed), the expected value of the next card adjusts accordingly, which is a key insight for dynamic pricing.
Clarify that we draw three cards without replacement from a standard 52-card deck, and we want the expected sum (fair value). Assume cards are drawn uniformly at random.
Calculate the expected value of a single card: sum of all card values divided by 52. For standard values (A=1, J=11, Q=12, K=13), this is 364/52 = 7. So expected sum of three cards is 3 * 7 = 21.
After each card is revealed, update the expected sum of the remaining cards by subtracting the revealed card's value from the total expected sum and adjusting for the reduced deck. Specifically, after revealing a card of value v, the new expected sum of the remaining two cards is (total expected sum of all remaining cards) / (remaining number of cards) * 2, but simpler: the expected value of the next card is the average of the remaining cards.
Walk through a concrete example: if the first card is a 10, the expected sum of the next two cards is (364 - 10)/51 * 2 = 354/51 * 2 ≈ 13.88, so total expected sum becomes 10 + 13.88 = 23.88. Show how this updates again after the second card.
Mention that this is a martingale property: the expected sum remains 21 before any cards are drawn, but conditional expectations change. Also note that for a data scientist role, this relates to online learning, Bayesian updating, and dynamic pricing.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Buy if your fair value is above the ask, sell if it's below the bid, pass if it's inside the spread.
Start by defining the two-sided market and the concepts of bid and ask, then explain that the decision depends on your estimate of fair value relative to the bid-ask spread and your risk tolerance. Emphasize that you should buy at the ask when your fair value is above the ask, sell at the bid when your fair value is below the bid, and pass when your fair value lies within the spread or when uncertainty is high.
Pro tip: Demonstrate awareness of adverse selection and inventory risk: even if your fair value suggests a trade, consider the probability that the counterparty has better information and the cost of holding inventory. A mature answer balances edge, risk, and market conditions.
Briefly explain what a two-sided market is and clarify the bid (price to sell) and ask (price to buy). Mention that the spread compensates market makers for risk and adverse selection.
Explain that your decision hinges on your estimate of the asset's fair value. This estimate should incorporate all available information, including order flow, news, and market conditions.
If fair value > ask, buying at the ask is profitable; if fair value < bid, selling at the bid is profitable. If fair value is between bid and ask, passing avoids a loss.
Adjust for risk aversion, inventory constraints, and adverse selection. Even if fair value suggests a trade, high uncertainty or unfavorable risk may warrant passing.
Recognize that conditions change: liquidity, volatility, and information asymmetry affect the decision. Be prepared to adapt your strategy in real-time.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
King is rank 13, so seeing it first pulls your expected sum up.
First, clarify the game setup and the original market price. Then, compute the updated fair value given the revealed King by considering the remaining deck composition and the distribution of the sum of three cards. Finally, compare the new fair value to the original market price to determine if the decision changes.
Pro tip: Show that you can quickly update probabilities using conditional expectation and that you understand the difference between fair value and market price, including transaction costs or edge.
Confirm the rules: three cards drawn without replacement from a standard deck, sum of values (Ace=1, face cards=10 or as specified), and the original market price for the sum.
Calculate the expected sum of three cards from a full deck to establish the baseline fair value.
Given the first card is a King, recompute the expected sum of the remaining two cards from the reduced deck, then add the King's value to get the new fair value.
Compare the updated fair value to the original market price. If the market price is unchanged, determine whether the new fair value implies a different action (buy/sell/hold).
Discuss how the decision might change if the market adjusts, or if there are multiple Kings, and mention the impact of card value conventions (e.g., Ace=1 vs 11).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This one stressed me out more than it should have.
Acknowledge the rule change, then systematically analyze how it alters optimal betting strategy by considering risk of ruin, expected value, and bankroll growth. Discuss how you would adapt your model or decision-making framework, emphasizing the trade-offs between aggressive betting and survival.
Pro tip: Show that you understand the Kelly criterion and how forced minimum bets can lead to over-betting, increasing risk of ruin. Quantify the impact if possible, and suggest dynamic adjustments to bet sizing based on bankroll fluctuations.
Restate the rule to ensure understanding: a minimum bet of 10% of current bankroll each round. Note that this is a forced minimum, which may exceed optimal bet sizes under certain conditions.
Compare the forced minimum to the optimal bet size from a model like Kelly criterion. Determine scenarios where the minimum forces over-betting, increasing risk of ruin.
Use simulations or analytical methods to estimate the probability of ruin and expected bankroll growth under the new constraint. Consider different win probabilities and payoffs.
Propose adjustments: e.g., become more conservative in other aspects, seek games with higher edge, or accept higher risk if forced. Discuss trade-offs between short-term survival and long-term growth.
Summarize how you would change your approach, emphasizing data-driven analysis and adaptability. Highlight any assumptions and potential limitations.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Spread should reflect your uncertainty about the fair value plus some profit margin.
Start by explaining that the spread should compensate for the risks you take as a market maker, primarily adverse selection and inventory risk. Then outline a practical method to estimate the fair value and the uncertainty around it, and set the spread width based on that uncertainty plus a profit margin. Finally, mention that you would adjust dynamically based on market conditions and your own risk appetite.
Pro tip: Emphasize that the spread is not just about transaction costs but about protecting yourself from informed traders; a good answer will quantify the adverse selection component using data or a simple model.
Clarify that you are quoting a two-sided market for the sum of three cards, meaning you must buy at your bid and sell at your ask. Your goal is to profit from the spread while managing the risk of being picked off by informed traders.
Calculate the expected value of the sum based on the distribution of cards (e.g., if cards are drawn from a standard deck without replacement). Assess the variance or uncertainty in this value due to information asymmetry or random fluctuations.
Break down the spread into: (a) adverse selection cost (expected loss to informed traders), (b) inventory risk premium (compensation for holding unwanted positions), and (c) profit margin. Use historical data or simulations to estimate these components.
Combine the components to set a bid-ask spread around the fair value. For example, if fair value is 20 and adverse selection is 0.5, inventory risk is 0.3, and desired profit is 0.2, the spread might be 2 (bid 19, ask 21). Adjust based on market conditions.
Continuously update your fair value and spread as new information arrives (e.g., cards revealed, order flow). Tighten the spread when competition is high or volatility is low, and widen it when uncertainty increases.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Acknowledge the positive edge but immediately pivot to risk of ruin and the difference between expected value and expected utility. Explain that even with a positive edge, betting the entire bankroll exposes you to catastrophic loss, and optimal bet sizing (e.g., Kelly criterion) balances growth and risk. Emphasize that in practice, edges are uncertain and markets can be adversarial.
Pro tip: Mention that at a firm like Optiver, risk management is paramount; even if you have an edge, you must consider the probability of drawdown and the need to survive to realize that edge. Show you understand that maximizing expected value is not the same as maximizing long-term wealth.
Clarify that a positive edge means the expected value of the bet is positive, but this does not guarantee a profit on any single bet.
Discuss that betting the entire bankroll leads to a non-zero probability of losing everything, after which you cannot recover or continue betting.
Mention the Kelly criterion or similar approaches that determine the fraction of bankroll to bet to maximize long-term growth while managing risk.
Highlight that in real-world situations, the edge is estimated and may be wrong; overbetting can lead to ruin if the true edge is smaller or negative.
Relate to trading or data science: even with a good model, position sizing and risk management are crucial to survive and profit in the long run.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Kelly maximizes long-run log growth under the assumption you know your edge precisely and repeat the same bet many times.
Start by defining the Kelly criterion and its goal of maximizing long-term logarithmic growth. Then discuss its assumptions (known probabilities, repeated bets, no constraints) and why they may fail in this specific game (e.g., estimation error, non-ergodicity, risk limits). Finally, propose an alternative sizing rule such as fractional Kelly, Bayesian approach, or risk-constrained optimization, and justify it with trade-offs.
Pro tip: Emphasize that in practice, overbetting due to parameter uncertainty is the biggest risk; fractional Kelly or a Bayesian shrinkage approach often outperforms full Kelly even when the model is correct. Also, mention that Optiver values pragmatic risk management over theoretical purity.
Explain that Kelly maximizes expected log wealth by betting a fraction of capital proportional to edge over odds. Mention formula f* = (bp - q)/b for simple bets.
List key assumptions: known true probabilities, infinite divisibility, no constraints, repeated independent bets. Note that real-world games often violate these.
Identify which assumptions fail: e.g., uncertain edge, non-stationarity, risk limits, or one-shot nature. Explain how these make full Kelly suboptimal or dangerous.
Suggest a practical alternative: fractional Kelly (e.g., half-Kelly), Bayesian Kelly with posterior sampling, or mean-variance optimization with constraints. Justify why it addresses the issues.
Compare alternatives in terms of growth, risk, and robustness. Mention how you would calibrate the rule (e.g., backtesting, simulation) and monitor performance.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.