Solve probability and game-theory puzzles

This is a quant trading internship interview. Work through the probability and game-theory puzzles below. You are not expected to respond immediately — pausing for a minute or two to reason or calculate before committing is normal. For each puzzle, lay out your strategy and reasoning clearly, and provide the requested numerical result whenever one is asked for.

The interviewer cares about how you think, not how quickly you retrieve a memorized fact. A clear argument — an upper bound, a posterior, a Bellman or stopping rule, an equilibrium idea — counts more than the final number.

Clarifying Questions to Ask

• For the bottle game: may I choose actions adaptively based on earlier removal-success feedback, or must I commit to a plan in advance? This determines whether the answer is a fixed split or a policy.
• For the coin game: is the bias $p$ fixed for the entire sequence, drawn once from the prior, or is it redrawn on every flip? Also, do I see each outcome before making the next guess?
• For the die game: when I stop, is my payout exactly the accumulated total $S$, and does rolling a 1 reset $S$ to $0$ with no later recovery?
• For the two-player die game: do I observe the opponent's running total or stopping decision, or are the two sequences fully independent and hidden until the end?
• For the dice-bidding game: are bids limited to strictly increasing integers, who opens the bidding, and is the payoff purely binary, meaning only win or lose the challenge?

What a Strong Answer Covers

Across the set, the interviewer is looking for these signals — name the principle, then compute:

• Upper and lower bounds before optimizing — can you cap what is achievable, for example with a counting argument, before searching for the optimal strategy?
• Bayesian updating — recognizing a conjugate prior, writing down the posterior, and using the posterior-predictive probability instead of a point estimate.
• Optimal stopping and dynamic programming — setting up a Bellman equation, finding the indifference threshold, and separating a one-step look-ahead from the full value function.
• Game theory under incomplete information — shifting the objective from maximizing your own expectation to maximizing your chance of winning, reasoning about what an opponent's action reveals, and using the idea of equilibrium, including when randomization or bluffing is required.
• Numerical follow-through — committing to a clean closed form or a defensible approximation, and sanity-checking it through limits, symmetry, or trivial baselines.

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1) Two bottles with add/remove actions (100 rounds)

You start with two bottles, A and B, both empty, and an unlimited supply of identical balls. The game runs for exactly 100 rounds. In each round you choose exactly one of these actions:

• Add — one ball is placed into a uniformly random bottle, either A or B. You do not see which bottle received it.
• Remove — one bottle is chosen uniformly at random. If that bottle is not empty, you remove one ball and observe that the removal succeeded. If it is empty, no ball is removed. The only feedback you ever receive is whether a removal attempt succeeded — you never learn which bottle was selected, and you never see the contents of either bottle. Goal: maximize the expected number of balls removed over the 100 rounds. Tasks:

1. Describe an optimal strategy.
2. Compute the expected number of balls removed under that strategy.

Hint — find the ceiling first. Before optimizing, bound what is possible. If the total rounds are split between adds and removes, what two simple counting limits constrain the maximum number of balls you could ever remove? Treat the split as a variable and ask where the cap is largest.

Hint — ordering matters. Once the split is fixed, consider whether adds and removes should be interleaved or separated. Which ordering tends to keep more inventory available before a removal attempt, and why might early removals waste attempts?

Hint — computing the expectation. Model the contents of one bottle after the adds, and the removal attempts that happen to target that same bottle, as two random quantities of the same kind. A failed attempt corresponds to an imbalance between attempts and inventory in a bottle — can you express total failures as a single absolute-difference quantity, then estimate its expectation?

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2) 100 coin flips with an unknown bias

A coin is flipped 100 times. The probability of Heads is an unknown parameter $p$, with prior $p \sim \mathrm{Unif}(0,1)$. Before each flip, you must guess Heads or Tails. You earn $1 for each correct guess and $0 for an incorrect one. You observe the outcome of every flip after guessing it, so your guesses may depend on the flips seen so far. Tasks:

1. What guessing strategy maximizes your expected total winnings?
2. Let $X$ be the upfront price to play. Treating yourself as risk-neutral, for what values of $X$ would you be willing to play?

Hint — conjugate prior. A $\mathrm{Unif}(0,1)$ prior is a special case of a $\mathrm{Beta}$ prior. After observing some heads and tails, what is the posterior over $p$, and what does it imply for the probability that the next flip is Heads? There is a classical named result for exactly this quantity.

Hint — what does per-flip scoring imply? Each guess earns $1 independently of the others, and you see the outcome before the next guess. Does maximizing the total expectation force you to plan ahead, or can it be decomposed flip by flip? Once you decide that, the per-flip decision follows from your posterior-predictive probability.

Hint — pricing. A risk-neutral player plays exactly when the price is below the expected payoff. So you need $\mathbb{E}[\text{winnings}]$, a sum or estimate over the 100 flips. What does the per-flip accuracy approach as data accumulates, and how does the whole thing compare with a trivial data-ignoring baseline such as always guessing Heads?

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3) Stop or continue with a 10-sided die, single-player and competitive

A fair 10-sided die with faces 1–10 is rolled repeatedly. You keep a running profit $S$, starting at $S = 0$. On each roll:

• Roll 2–10: that value is added to $S$.
• Roll 1: the game ends immediately and your final payout is $0, because your accumulated $S$ is wiped out. Before each roll, while the game is still going, you may instead choose to stop and take your current $S$ as your payout. Tasks — single-player: Determine the optimal stopping rule.

Hint — one-step look-ahead. Compare stopping now with rolling exactly once more and then stopping. Each continued roll has a known bust probability and, when it does not bust, adds a value with a known expectation. Write the value of rolling once in terms of your current total, and ask when it beats stopping.

Hint — why the threshold is exact. An extra roll trades a small chance of destroying your entire accumulated total against a likely gain. Set the expected gain from continuing equal to the expected loss to locate the indifference point. A full Bellman or backward-induction argument should confirm that this one-step boundary is the true stopping boundary — can you see why no extra option value shifts it here?

Tasks — two-player competition: Two players each play this same game with their own independent die sequence, and each chooses when to stop. The player with the higher final payout is paid their own final payout by a third party; the loser receives $0. If the payouts tie, they split equally. Describe how optimal play changes relative to the single-player case.

Hint — the objective changed. You are no longer maximizing $\mathbb{E}[S]$; you now care about finishing ahead of the opponent's final-score distribution. Re-derive what you are actually optimizing, and ask whether maximizing your mean and maximizing your chance of outscoring the opponent point the same way or pull your threshold in different directions.

Hint — do not expect a clean pure threshold. Treat this as a game: best-respond to an assumed opponent threshold, then best-respond to that, and watch what the iteration does. Does it converge to one shared number, or not? Whatever you conclude about a single closed-form answer, the most defensible deliverable is the direction the competition pushes you relative to the single-player rule.

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4) Bidding on the sum of two dice with private information

Two fair six-sided dice are rolled, one assigned to each player.

• You see only your own die outcome $a$.
• Your opponent sees only their own die outcome $b$. Players alternate turns in an ascending-bid game about the total sum $a + b$:
• On your turn, you either raise to a strictly higher integer bid $k \in \{2, \dots, 12\}$, or challenge the previous bid.
• A bid $k$ is a claim that the true sum satisfies $a + b \ge k$.
• When a player challenges, both dice are revealed:
  • If $a + b \ge k$, the last bidder, the one who made bid $k$, wins.
  • Otherwise, the challenger wins. Task: Describe an optimal strategy — how to decide whether to raise or challenge, and what value to raise to — as a function of your observed die value $a$ and the bid history.

Hint — your posterior over the sum. You know $a$; the opponent's $b$ is uniform on $\{1,\dots,6\}$. From your seat, what is the distribution of the sum, and hence $\Pr(a+b \ge k \mid a)$ as a function of $k$? Identify the two extreme regimes of $k$ where the claim becomes certainly true or certainly false given your die — those bound what you can safely support.

Hint — what does a binary payoff reward? The payoff is purely win or lose, not proportional to the margin. For such a payoff, what is the single comparison that decides whether an event is worth betting on at any node? Frame both challenging the current bid and raising to a new bid in terms of that comparison applied to your success probability.

Hint — what does a raise tell you? Ask why a rational opponent would voluntarily raise to a high number — what does that reveal about their hidden die? Should you keep using the flat prior on $b$ after seeing the bid history, or condition on it? Then turn it around: your own bids leak information too. What does that symmetry imply about whether a purely honest, fully-revealing strategy can be safe against a competent opponent?

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Follow-up Questions

After your main answers, expect deeper probes such as:

• Bottle game: how does the optimal split and expected count change if the game lasts $2N$ rounds instead of 100? What if removal feedback, success or failure, could actually be used adaptively — does it help?
• Coin game: how does the expected payoff change for a general $n$ flips, and what is the per-flip accuracy in the limit of large $n$? How would a risk-averse player adjust the price they would pay?
• Die game: if you could observe the opponent's already-locked final score before finishing, what is the optimal rule? How does the two-player threshold respond as the opponent plays more or less aggressively?
• Dice bidding: roughly how much bluffing is required at equilibrium, and why does always-honest bidding lose to a competent opponent? How does the analysis extend if each player rolls two dice instead of one?

Overview: This set of puzzles evaluates probabilistic reasoning, Bayesian inference, expected-value optimization, optimal stopping, and strategic game-theoretic reasoning under uncertainty and asymmetric information, in the Statistics & Math domain.