← Jane Street Interview Insights
This part felt easy and I almost rushed past it.
Clarify that the game stops as soon as one player reaches 4 wins, so the maximum number of rounds is the sum of the winner's 4 wins and the loser's maximum possible wins without reaching 4, which is 3. Thus, the maximum is 7 rounds. Explain that this is a worst-case scenario where the score is 4-3.
Pro tip: Emphasize that the coin being fair is irrelevant to the maximum; it only affects probabilities. This shows you can separate structural constraints from probabilistic ones.
The game stops immediately when either player reaches 4 wins. So the winner must have exactly 4 wins, and the loser can have at most 3 wins.
The maximum number of rounds occurs when the loser has as many wins as possible without reaching 4, i.e., 3 wins. Thus, total rounds = 4 + 3 = 7.
Consider a sequence where Player 1 wins 4 and Player 2 wins 3, with the final win for Player 1 occurring in the 7th round. This is possible, so 7 is achievable.
Clarify that the game cannot last 8 rounds because that would require the loser to have 4 wins, which would have ended the game earlier. Also note that the fairness of the coin does not change the maximum.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the game rules: assume a fair coin and that the game ends as soon as one player wins 4 flips (best-of-7). The 7th flip is needed if and only if after 6 flips the score is tied 3-3. Compute the probability of a 3-3 tie after 6 flips using the binomial distribution, then multiply by 1/2 for the 7th flip to be decisive.
Pro tip: State your assumptions explicitly (fair coin, independent flips, game stops when someone reaches 4 wins) and then simplify the problem by recognizing that the 7th flip is required exactly when the first 6 flips result in a 3-3 split. This shows structured thinking and avoids overcomplicating the calculation.
Confirm that the game is a best-of-7 series where the first player to win 4 flips wins, and that flips are independent and fair. This ensures the problem is well-defined.
The 7th flip is needed if and only if after 6 flips the score is tied 3-3. If either player has 4 or more wins after 6 flips, the game would have already ended.
Model the first 6 flips as a binomial experiment with n=6 and p=0.5. The probability of exactly 3 successes (e.g., heads) is C(6,3) * (0.5)^6 = 20/64 = 5/16.
Given a 3-3 tie, the 7th flip always determines the winner (since one player will reach 4 wins). Thus, the probability that the game requires all 7 rounds is simply the probability of a 3-3 tie after 6 flips, which is 5/16.
State the probability as 5/16 or 31.25%. Optionally, discuss the intuition: it's less than 1/2 because ties after 6 flips are not the most likely outcome.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
I spent way too long on this trying to set up a Bayes calculation.
First, clarify the game rules and the meaning of 'Player 1 won' (e.g., most heads after 7 rounds). Then, use conditional probability: given the first three rounds have exactly 2 heads and 1 tail, the remaining four rounds are independent, so compute the probability that Player 1 wins based on the total heads needed. Finally, sum over the possible outcomes of the remaining rounds that lead to a Player 1 win.
Pro tip: State your assumptions explicitly (e.g., fair coin, win condition) and then solve the problem systematically. This shows you can handle ambiguity and communicate clearly, which is crucial in interviews.
Ask or state the rules: Is it a fair coin? What does 'Player 1 won' mean? Typically, Player 1 wins if they have more heads than Player 2 after 7 rounds. Confirm this to avoid misinterpretation.
Let H be the total number of heads in 7 rounds. Player 1 wins if H > 3.5, i.e., H ≥ 4. Given the first 3 rounds have exactly 2 heads, the remaining 4 rounds determine the total.
Let X be the number of heads in the remaining 4 rounds. X ~ Binomial(4, 0.5). Player 1 wins if 2 + X ≥ 4, so X ≥ 2. Compute P(X ≥ 2) = 1 - P(X=0) - P(X=1) = 1 - (1/16 + 4/16) = 11/16.
Double-check the calculation and state the final probability as 11/16 or 0.6875. If the win condition is different (e.g., Player 1 wins if total heads is odd), adjust accordingly.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the game rules: two independent sequences of fair coin flips, Player 1 wins when they have n heads, Player 2 wins when they have n tails. The maximum game length occurs when the game is decided on the very last flip, meaning after 2n-1 flips the score is (n-1) heads and n tails or n heads and (n-1) tails, and the final flip gives the winner their nth required outcome. Compute the probability of this event using the negative binomial distribution or combinatorial counting, then generalize for any n.
Pro tip: Emphasize that the game must end by flip 2n-1 because by then one player must have at least n of their target outcome (pigeonhole principle). Also, note that the probability of reaching the final round is the sum of two negative binomial probabilities, which simplifies to a clean expression.
Restate the rules: Player 1 needs n heads, Player 2 needs n tails, flips are independent fair coins. The game stops as soon as either player reaches their goal. Determine the maximum possible number of flips.
Use the pigeonhole principle: after 2n-1 flips, one player must have at least n of their target outcome, so the game cannot exceed 2n-1 flips. The maximum length is 2n-1.
The game reaches the final round (flip 2n-1) if and only if after 2n-2 flips, the score is (n-1) heads and (n-1) tails. Then the next flip decides the winner.
The probability of having exactly n-1 heads and n-1 tails in 2n-2 flips is C(2n-2, n-1) / 2^{2n-2}. Since the final flip is irrelevant to reaching the final round, this is the desired probability.
Check for small n (e.g., n=1 gives probability 1, n=2 gives 3/8) and express the result in terms of binomial coefficients. Discuss asymptotic behavior if relevant.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
With a biased coin the 7th flip still decides everything, so P(Player 1 wins | game lasted 7 rounds) = p.
First, clarify the game rules and what 'Player 1 won' means in the context of a biased coin and a fixed number of rounds. Then, use conditional probability and symmetry arguments to determine whether the first three rounds affect the outcome, considering the stopping condition and the bias parameter p.
Pro tip: Demonstrate that you can separate the effect of the stopping rule from the coin's bias, and use symmetry to simplify the problem. This shows deep understanding of probability and avoids unnecessary calculations.
Ask clarifying questions to understand the rules: What defines a win? Is the game stopped after 7 rounds? What is the role of the first three rounds? This ensures you address the correct problem.
Determine the sample space of outcomes given the stopping condition. Recognize that conditioning on the game lasting exactly 7 rounds may restrict the possible sequences.
Use symmetry between players if the game is fair in terms of rules, and compute the conditional probability of Player 1 winning given the game lasted 7 rounds. Consider whether the first three rounds provide additional information beyond the stopping condition.
Determine if knowing the composition of the first three rounds changes the conditional probability. This may involve checking if the event of lasting 7 rounds is independent of the early rounds given the bias.
State the final probability and whether the first three rounds matter. Discuss how the result depends on p and the game rules, and mention any assumptions made.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Ran out of time on this and just said it requires summing over all possible game lengths weighted by their probabilities.
First, clarify the rules of the game and define the state space and transitions. Then, set up equations for the expected number of rounds from each state, often using first-step analysis or Markov chain techniques, and solve them to get the final answer.
Pro tip: Always state your assumptions and define the stopping condition clearly; interviewers value a structured, transparent thought process over a rushed numerical answer.
Ask questions to confirm the exact rules, such as what constitutes a round, when the game ends, and any special conditions. This ensures you solve the correct problem.
Identify the relevant states (e.g., current streak length, score) and the probabilities of moving between them. For a fair coin, each transition has probability 1/2.
Let E_i be the expected number of rounds from state i. Use first-step analysis: E_i = 1 + sum over next states j of P(i->j) * E_j, with E_terminal = 0.
Solve the system of linear equations for the initial state. If the state space is large, look for patterns or use generating functions.
Check the solution for reasonableness (e.g., non-negative, finite) and relate it back to the game. Consider edge cases or alternative methods to confirm.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.