Dice-Based Trading Games

A standard interview format: a die is rolled, and you have to quote a two-sided market on the outcome, either before the roll or after seeing partial information (a second die, a reveal that the result is even, and so on). It sounds like arithmetic, but the real test is whether you correctly recompute the expected value every time the available information changes, under time pressure, without dropping a term.

The building block: a fair die's expected value

A single fair six-sided die has expected value $E[X] = \frac{1+2+3+4+5+6}{6} = 3.5$ — the average of all six equally likely outcomes. This isn't a possible outcome of any single roll (you can't roll 3.5), which is itself a useful thing to say out loud in an interview: expected value is a long-run average, not a prediction of any one roll.

Worked example: quoting after a partial reveal

You're asked to quote a market on the sum of two fair dice, but before you quote, the interviewer tells you the first die landed on an odd number (1, 3, or 5, each still equally likely). Recompute from scratch rather than adjusting the unconditional answer by feel: the first die's conditional expected value is $\frac{1+3+5}{3} = 3$, and the second die is unaffected, still $3.5$. The sum's expected value is $3 + 3.5 = 6.5$ — noticeably below the unconditional two-dice sum of $2 \times 3.5 = 7$, because you've ruled out the highest even values (2, 4, 6) on the first die. A candidate who just eyeballs "7, roughly" and doesn't redo the calculation after the reveal has missed the entire point of the question — the reveal exists specifically to test whether you re-derive the number rather than reuse it.

Now suppose instead you're told the sum of the two dice is known to be even, with no other information. Every combination that's still possible (both odd, or both even) is equally weighted among the valid pairs, and by symmetry around the centre value the conditional expectation stays at 7 — a useful check that not every piece of revealed information should move your price, and part of the test is recognizing which ones do.

What this means in practice

Dice games are a proxy for a real desk skill: fast, correct conditional expectation under a stream of partial signals — an order that reveals only its side, a headline that rules out some scenarios but not others. The mechanics are identical to conditioning on a die's parity; the value is training the reflex to redo the sum every time new information narrows the outcome space, rather than nudging an old number.