Graham and Andrea Work Through a Trading Mock Interview

Michael introduces a trading mock interview: Graham, an experienced Jane Street trader, asks newer trader Andrea a problem previously used in real interviews. They explain that interviews often involve working with an interviewer on a challenging quantitative problem. Andrea has not seen this question before; it is not currently in use.

Graham describes a one-player game with a 20-sided die, initially showing 1, and 100 rounds. On each round the player either rolls, replacing the current face, or takes that many dollars from the casino. The goal is to decide how to play and estimate average earnings.

Andrea asks whether rolls are limited or cost money. A roll consumes a round but has no other charge. She initially thinks collecting money ends the game; Graham clarifies that play continues through all 100 rounds and money can be collected repeatedly.

Andrea asks what happens to the die after collecting. Graham gives an example: if a 17 is showing and she takes $17, it remains on the table for the next round. She can collect $17 again or choose to reroll. Working through the concrete sequence helps her establish the rules.

Andrea considers rolling until a 20 appears, then collecting 20 for the remaining rounds. A 20 takes about 20 rolls on average. She initially counts money earned during those rolls, but Graham reminds her that rolling and collecting are alternatives; her rough estimate becomes $1,600 for the remaining 80 rounds.

If she accepts either 19 or 20, the wait averages about 10 rolls and the accepted face averages 19.5. Multiplying by roughly 90 remaining rounds gives about $1,755, better than the 20-only estimate. These are rough comparisons, not an exact finite-horizon calculation.

Andrea tries accepting 18, 19 or 20: the chance per roll is 3/20, the wait is a little under seven rolls, and the accepted face averages 19. Her rough estimate is about $1,767. Accepting 17 through 20 takes about five rolls, but the lower average of 18.5 gives roughly $1,758.

On those estimates, Andrea favors rolling until 18 or higher, then collecting repeatedly. Graham asks where the average of 19 came from; she explains that 18, 19 and 20 are equally likely when one of them is accepted.

Graham changes the game: after the player collects, the die is removed, so a new roll must consume a round before another collection. Andrea initially proposes accepting lower values, but asks for clarification about how collecting and rolling alternate.

Graham explains that roll, collect, roll, collect consumes four turns; alternatively, the player can roll more than once before collecting. Andrea now compares always collecting the next face with accepting only values from 11 to 20.

Accepting the top half takes about two rolls on average, followed by a collection turn. Its accepted value averages 15.5, so roughly 33 cycles in 100 turns yield about $511. Always rolling once and collecting takes two turns per cycle, with an average face of 10.5, for about $525.

Andrea generalizes to accepting the top fraction x of faces. She estimates 1/x rolls plus one collection turn per cycle, giving roughly 100x/(1+x) cycles. Using an approximate accepted-value expression, 20 − 10x, she obtains an estimated total of 1000x(2 − x)/(1 + x).

Rather than do the algebra during the interview, Andrea says she would differentiate, set the derivative to zero and check that the result is a maximum and better than her baseline. Graham says the threshold works out to taking 6 or higher. He offers an intuition: if earnings average about $5 per turn, spending a turn to collect $7 is attractive.

In the third variant the face can remain after collection, as in the first game, but the casino may reroll it once after each collection without costing the player a turn. The casino wants to minimize the player’s earnings. Graham invites Andrea to reason about the game even if they cannot solve it exactly.

Andrea confirms that the casino may reroll only after she collects, and that if it does not, the same face remains available. She imagines the player and casino each using a threshold, potentially settling at a face neither side wants to change.

An 18-or-higher strategy would prompt the casino to reroll high results, frustrating the player’s plan to keep them. Andrea suggests lowering the threshold and guesses an equilibrium near 10 or 11, where neither side has much incentive to reroll. Graham asks her to choose a concrete threshold and play out the consequences.

They consider the player collecting at 10 or higher while the casino rerolls 11 or higher. To simplify the calculation, Andrea also considers a common threshold of 11: values above it benefit the player, so the casino rerolls after collection; lower faces lead the player to roll.

Andrea initially imagines the casino rerolling high faces immediately, but Graham corrects her: it gets that option only after a collection. For an 11-or-higher threshold she estimates a 15.5 average accepted value, about 50 such collections and roughly $775 over the game.

Graham asks for the return per turn. At roughly $7–8 per turn, Andrea realizes that taking an 8 might be worthwhile. They test a threshold of 8 through 20, whose accepted face averages 14 and whose chance per roll is 13/20.

Andrea initially adds a collection turn to the expected wait. Graham points out that the casino’s reroll after collecting gives the player a new random face for free, so the cycle differs from the second version. Using approximately 100/(20/13) = 65 collections at $14 each, they estimate $910. They leave the precise best strategy unresolved.

Graham ends the mock interview, noting that a real interview might contain more questions and time for the candidate to ask about Jane Street or an internship. In the recap he says Andrea did well but reminds viewers that she already has experience as a trader and with problems like this.

It took them a little while to agree on the first game’s rules. Graham says that is fine: Andrea kept clarifying until she understood, avoiding a lot of work on the wrong problem. He also values her thinking aloud, which showed her reasoning and helped him decide when to hint or correct a misunderstanding.

Andrea remembers being encouraged to think aloud in her own interviews; it became more natural with practice. Graham says candidates are not expected to talk quite as much as she did, but communicating their thinking is useful.

Graham also praises Andrea for breaking the quantitative problem into smaller pieces and testing strategies even when they might not be optimal. This iterative, collaborative approach resembles conversations on a trading desk, and it allowed her to use hints productively.

Andrea recalls being nervous before her own early interviews, but says her interviewers put her at ease and made them feel like exciting discussions. They close by wishing viewers luck with their interviews.