Say I hand you a coin that lands heads 60% of the time, put $100 in front of you, and let you bet any fraction of your money on heads at even odds, a hundred times. Heads pays you the amount you staked, tails takes it. How much do you bet on each flip? Most people, including quick ones, either bet a flat dollar amount or shove far too much, and a surprising number find a way to go broke holding a coin that’s rigged in their favor.
That last part isn’t a hypothetical. In 2016 Victor Haghani, one of the founders of LTCM, ran this exact setup with finance students and young professionals: a coin they were told came up heads 60% of the time, real money, thirty minutes to bet. Roughly a third walked away with less than they started, and only about one in five bet their way to the cap. People who trade for a living stared at a favorable coin and lost money on it.
The two instinctive answers are both wrong in instructive ways. Betting everything on each flip maximizes what you expect to hold after a single flip, but string a hundred flips together and a tails is a near-certainty, so “bet it all” is a guarantee of ruin dressed up as expected-value maximization. Flat betting, say five dollars every time regardless of your balance, never blows up but leaves most of the compounding on the table. The right answer is to bet a fixed fraction of whatever you currently hold, and that fraction has a name.
The Kelly fraction is edge over odds
For a bet that pays b to 1 when you win with probability p (and q = 1 - p when you lose your stake), the growth-optimal fraction of your bankroll to wager is
f* = (bp - q) / b = p - q/bAt even money, b = 1, so it collapses to f* = p - q = 2p - 1, which is just your edge. A 60/40 coin gives f* = 0.2. Bet 20% of your current pile on every flip, resizing as the pile grows and shrinks, and you’re doing the mathematically best thing available. If the coin instead paid 2 to 1 on a win, the same 60% edge would justify f* = 0.6 - 0.4/2 = 0.4, a much bigger bet, because better odds on the upside let you press harder.
Notice the fraction is of your current wealth, not your starting stake. After a few heads you’re betting more dollars; after a tails you’re betting fewer. That self-correction is the whole point, and candidates who quote a fixed dollar figure have already missed it.
Maximizing growth, not expected value
The reason a fraction beats going all-in is that money compounds multiplicatively. Your wealth after a hundred flips is a product of a hundred terms, not a sum, so the quantity that governs your long-run outcome is the expected logarithm of wealth, which is the geometric growth rate per flip:
g(f) = p * ln(1 + b*f) + q * ln(1 - f)Differentiate that with respect to f, set it to zero, and the Kelly fraction falls out. No utility function required. You don’t have to assume you’re risk-averse or that you value dollars on a log scale as a matter of taste. It’s simpler than that: over enough repeated bets, the fraction that maximizes g(f) produces more wealth than any other fixed fraction with probability approaching one. Betting to maximize expected wealth per flip and betting to maximize the growth rate of wealth are different objectives, and the interview is checking whether you know which one survives compounding.
This is why the biased-coin question is a favorite. It looks like an expected-value puzzle, and the expected-value answer (bet everything, since each flip has positive expectation) is exactly the trap. Expected value is linear and blind to ruin. Growth rate is not.
What over-betting costs you
Plot g(f) for the 60/40 coin and you get a hump. Growth climbs from zero, peaks at the Kelly fraction, then falls back down and crosses zero at precisely twice Kelly. Bet more than that and your growth rate goes negative: you own an edge and still bleed money, because the volatility drag from over-sizing overwhelms it. The table shows what that looks like over a hundred flips starting from $100.
| Fraction of bankroll bet per flip | Expected log-growth per flip | Typical bankroll after 100 flips (from $100) |
|---|---|---|
| 0% (never bet) | 0.0000 | $100 |
| 10% (half Kelly) | 0.0150 | $450 |
| 20% (Kelly) | 0.0201 | $749 |
| 25% | 0.0188 | $655 |
| 30% | 0.0147 | $437 |
| 40% (twice Kelly) | -0.0024 | $78 |
| 50% | -0.0340 | $3 |
Read the asymmetry off that table, because it’s the part interviewers actually care about. Under-betting at 10% still grows your money nicely, just slower than optimal. Over-betting at 40%, double the Kelly fraction, turns a rigged-in-your-favor coin into a losing proposition, and at 50% you’re all but wiped out. The cost of being too timid is small; the cost of being too aggressive is catastrophic. That lopsidedness is why the second half of any good version of this question is about what you don’t know.
Half Kelly, and why real desks bet less
Almost nobody trading actual capital bets full Kelly. Two reasons, and a strong candidate names both.
The first is variance. Full Kelly is a wild ride: the growth rate is maximized, but the path there swings hard, and a Kelly bettor has a genuine chance of seeing their bankroll cut in half before it doubles. Because the growth curve is flat near its peak, backing off to half Kelly costs you only about a quarter of your growth rate while cutting the variance of your wealth roughly in half. Most people will trade a little growth for a lot less stomach-churn, so half or even quarter Kelly is the working default.
The second reason is the one that matters more, and it’s the tell that separates people who memorized the formula from people who understand it. You never actually know p. The coin in the puzzle is stated as 60/40; a real edge is estimated, noisy, and drifting. If you believe the edge is 60/40 but it’s truly 55/45, then full Kelly on your estimate is over-betting on reality, and over-betting is the dangerous side of the curve. Shading your bet size below what your estimate suggests is insurance against your own measurement error. When an interviewer follows up with “what if you’re not sure it’s exactly 60%,” the answer they want is that uncertainty in p pushes you toward a smaller fraction, not a larger one.
From one coin to a book of bets
The coin is a warm-up. The version that shows up in a portfolio or trading-desk round replaces the discrete bet with a continuous return. For an asset with expected excess return μ over the risk-free rate and variance σ², the growth-optimal amount of leverage is
f* = μ / σ²and the growth rate you achieve at that sizing works out to the risk-free rate plus half the Sharpe ratio squared. That connection is worth saying out loud in an interview, because it ties bet sizing straight to the Sharpe ratio the desk already lives by: doubling your Sharpe quadruples your optimal growth rate, and the best fraction to bet is a direct function of return-over-variance, not return alone. A high expected return on something violently volatile deserves a small position.
Push it one more step and you’re sizing several bets at once. When positions are correlated, you can’t Kelly each one in isolation, because their combined variance depends on how they move together. The multi-asset Kelly solution is f* = Σ⁻¹μ, the inverse covariance matrix times the vector of expected returns, which is the same object mean-variance optimization spits out. The practical lesson interviewers are probing for is that two positively correlated “independent” edges are really one larger bet, and sizing them as if they were separate quietly doubles your risk.
How it shows up in the room
Bet-sizing questions are a staple at the options and market-making shops, SIG, Jane Street, Optiver, Akuna, Five Rings, and they rarely stop at the first answer. The opener is usually the biased coin or a dice game with a stated edge, and the interviewer wants the fraction and the reasoning behind it, not a memorized 2p - 1. Then come the variations:
- What if a win pays 2 to 1 instead of even money?
- What if you have to commit dollars up front and can’t resize as you go?
- What if you only get ten flips instead of a thousand?
- What if you’re not certain the coin is 60/40?
- How would you size two bets that are 70% correlated?
The failure modes are predictable. Someone maximizes expected value and bets everything. Someone quotes a flat dollar stake and can’t say why the answer should scale with current wealth. Someone recites the formula but can’t explain why over-betting is worse than under-betting, or why real traders shade below Kelly. Getting the number is table stakes; the hire signal is whether you can reason about the shape of that growth curve and where your own ignorance sits on it.
Which is the thing to carry out of the puzzle and into a real position. The coin might genuinely be 60/40, but your knowledge of it never is, and Kelly quietly assumes you know the edge exactly. You’re always betting a fraction of an estimate, so the sensible move is to bet a fraction of the fraction.
