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How professional gamblers size bets

Feb 2026

You have $100; and I offer you a steal of a deal. You get to bet using a biased coin that lands heads 60% of the time. Each flip, you bet whatever you like: heads wins your bet, tails loses it. Simple.

Try it.1

1: A 60/40 coin flip game. Reach $350 to win.
2: Your running balance across every game.

It’s surprisingly easy to go broke. Haghani and Dewey ran this experiment on 61 quantitative finance students — people who want to calculate expected value for a living. Yet 30% of them went bust on a 60/40 coin - yikes.2

If you bet small, hold that thought; if you went bust, you’re in good company - I did!

You knew this instinctively

We instinctively resist betting everything, even on good odds. Behavioural economists call this loss aversion — the finding that losses loom twice as large as the equivalent gains.3

The standard interpretation is that we are not particularly rational. However, I’d encourage you to take a moment to think about compounding losses. When outcomes compound — and in life, they almost always do — big losses really are worse. Going from $10,000 to $2,000 is catastrophic, because it takes as long to grow $2,000 to $10,000 as it takes to go from $10,000 to $50,000. You didn’t only lose $8,000; it also cost you your next $40,000.4

Loss aversion might not be a bias at all. It might be a rough-and-ready heuristic for the geometric expectation.

Consider laptop insurance. You pay £8 a month to cover a £1,200 machine. By expected value, the insurer makes money and you lose money. But think about it geometrically. If you’re a student and your laptop is worth a month of living expenses, losing it without cover is a 40% drawdown on your savings. For the insurer, your claim is one of fifty thousand - it definitely doesn’t matter to them. While it will matter to you. 5

3: Enter your premium, potential loss, and savings to compare arithmetic and geometric expected value. A bad deal by expected value can be a good deal geometrically.

This is why you buy insurance despite the “bad odds”, why you overpay for certainty in a dozen small ways that expected-value would call irrational. You actually have great intuitions for geometry.

Most people still fail

So you know to hold back - but you should probably still take some risk to win. Can we quantify how much?

Expected value — the standard answer — tells you what happens on average, across many people playing once. It tells you nothing about what happens to you, playing many times.

Rory Sutherland puts it well: despite your schooling, 10 × 1 ≠ 1 × 10.6

Two scenarios:

  1. 100 people each bet 40% of their $100 balance once. You collect the combined winnings.
  2. You bet 40% of your balance, 100 times in a row.

Scenario one ends you with roughly the expected value (you make money 97% of the time). Scenario two is where things are less cheery (you lose money 54% of the time). Why? If you win 40% then lose 40%, it doesn’t mean you break even. You’re down to 84% of where you started.

This distinction has a name: ergodicity. A system is ergodic when the average across a population equals the average across time for a single participant.

Life is not ergodic.

Nobody would play Russian roulette five times at £100 million a pull, even though the expected payout is about £200 million.7 The ensemble average (across all possible outcomes) says you’d expect to survive and be rich. The time average (your actual experience) says you’re probably dead.

4: 100 players bank £100M for every pull of a six-chamber revolver they survive. The ensemble average is a fortune; the typical player is dead.

The Kelly Criterion

John Kelly, a researcher at Bell Labs, solved this in 1956.8 The question isn’t “should I bet?” but “how much?”

For a biased coin at even money, the formula9 is:

f∗=2p−1f^* = 2p - 1

where pp is the probability of winning. At 60%, that’s f∗=0.2f^* = 0.2 — bet 20% of your balance.

Kelly maximises the expected log growth rate of your wealth.10 Expected value says bet everything. Kelly says bet enough that the geometric growth rate is highest.

Kelly did get pipped to first place. Daniel Bernoulli11 argued in 1738 that the value of a sum of money is not its face value but its logarithm — that gaining £100 matters more when you have £200 than when you have £20,000.12

5: Expected log-growth rate by bet fraction for a 60/40 coin at even money. The peak — the Kelly fraction — sits at f* = 20%.

At the Kelly fraction, growth is maximised. Bet 100% and you’re guaranteed to go bust eventually.

6: What happens at double Kelly?
Answer the quiz above to reveal

Try again with the Kelly marker visible.

7: The same game, now with the Kelly fraction marked in green.

Red Button, Green Button

A thought experiment. Someone offers you a one-time choice:

Neither button risks your existing wealth.14

8: Poll results and breakeven wealth for the red-button gamble.

Full Kelly says the breakeven is roughly $10,000 — if you have more than that, the red button is a better choice than the green. If that feels absurdly low, good. Full Kelly is quite risk-tolerant. It corresponds to a risk aversion parameter (γ\gamma) of 1. Raise γ\gamma and the breakeven rises with it:

γ\gammaKelly fractionBreakeven
1Full Kelly~$10K
2Half Kelly~$1M
3Third Kelly~$2.4M
4Quarter Kelly~$3.9M

Most empirical estimates of human risk aversion sit at γ\gamma = 2–4. Most practitioners bet half to quarter Kelly.15 Both land in the same range: $1M–$4M.

You can decide your own γ\gamma.

Stay in the Game

The Kelly criterion gives you a number, but the principle is simpler: stay in the game.16 Size your bets so that no single loss can knock you out. If you can split the wager across independent bets, so much the better; two losing bets can even combine into a winning one. The best strategy you never execute because you went bust on round three is worse than a mediocre strategy you can sustain for a thousand rounds.

The coin is biased in your favour. Just don’t bet the lot.

Section titled Footnotes

Footnotes

  1. The coin flip simulation is shamelessly stolen from heavily inspired by Unfair Flips, a delightful $2 game about flipping a coin that hates you, by Heather Flowers. ↩

  2. Haghani, V. & Dewey, R. (2016). “Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin.” Students were given $25 and a 60/40 coin. 28% reached the $250 cap. The median player left with less than they started with. ↩

  3. Kahneman, D. & Tversky, A. (1979). “Prospect Theory: An Analysis of Decision under Risk.” Econometrica, 47(2), 263–291. ↩

  4. The 10k→10k → 2k example is adapted from “The Misunderstood Kelly Criterion” at Entropic Thoughts. ↩

  5. A £1,200 loss is a 0.001% drawdown for an insurer with £100 million in reserves; it’s a 40% drawdown for a student with £3,000 in savings. ↩

  6. Rory Sutherland, Alchemy: The Surprising Power of Ideas That Don’t Make Sense (2019). ↩

  7. Standard six-chamber revolver, £100M banked for each pull you survive — but die and your winnings die with you. P(surviving all five pulls) = (5/6)⁵ ≈ 40%, so the expected payout is 0.40 × £500M ≈ £200M. You’re more likely to die than survive, but the expected payout is still a fortune. ↩

  8. Kelly, J.L. Jr. (1956). “A New Interpretation of Information Rate.” Bell System Technical Journal, 35(4), 917–926. ↩

  9. The general formula f∗=(bp−q)/bf^* = (bp - q)/b reduces to f∗=2p−1f^* = 2p - 1 when b=1b = 1 (even money). ↩

  10. Two pieces prompted this essay: “The Misunderstood Kelly Criterion” at Entropic Thoughts and Paul Butler’s interactive explainer. I wanted to explore what neither quite covers — why our gut instincts about risk might be doing the right calculation all along. ↩

  11. Great name - and yes, that Bernoulli. ↩

  12. Bernoulli, D. (1738). “Specimen Theoriae Novae de Mensura Sortis” (Exposition of a New Theory on the Measurement of Risk). Commentarii Academiae Scientiarum Imperialis Petropolitanae, 5, 175–192. Translated by Louise Sommer in Econometrica (1954). Bernoulli proposed that rational decision-making should maximise expected utility rather than expected value (utility = log wealth). Bernoulli was solving for how to evaluate a bet; Kelly was solving for how much to wager. ↩

  13. Exactly true in the continuous (log-normal) case, where the growth rate is g(f)=fμ−f2σ2/2g(f) = f\mu - f^2\sigma^2/2 and double Kelly perfectly cancels. For discrete binary bets it’s an approximation: the actual growth rate at f=0.4f = 0.4 on a 60/40 coin is 0.6ln⁡(1.4)+0.4ln⁡(0.6)≈−0.0020.6\ln(1.4) + 0.4\ln(0.6) \approx -0.002, slightly negative rather than exactly zero. Close enough for the intuition; the zero crossing is at f≈0.39f \approx 0.39. The growth-rate chart plots the continuous approximation, so its curve crosses zero at exactly 2f∗2f^*. ↩

  14. Wealth here means liquid net worth: cash, savings, and investments you could sell within a week. It excludes your house, pension, and future salary. ↩

  15. Half Kelly is the pragmatist’s choice: you sacrifice some growth for much lower variance and a near-zero chance of ruin. ↩

  16. Nassim Nicholas Taleb has been saying this louder and at greater length for decades. His first rule of risk is not “maximise returns” but “don’t go bust”. See Fooled by Randomness (2001) and Skin in the Game (2018). ↩