How professional gamblers size bets

The Game

You have $100. A biased coin lands heads 60% of the time. Each flip, you bet whatever you like: heads wins your bet, tails loses it. Simple.

Try it.¹

If you’re anything like most people, you went broke. Despite a coin that favours you. Despite positive expected value on every single flip.

You’re in good company. Haghani and Dewey ran this experiment with 61 quantitative finance students — people who want to calculate expected value for a living. 30% of them went bust on a 60/40 coin.²

Why Most People Fail

Expected value 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: 10 × 1 ≠ 1 × 10.³ In real life, multiplication is not commutative.

Two scenarios:

1. 500 people each bet 40% of their $100 balance once. You collect the combined winnings.
2. You bet 40% of your balance, 500 times in a row.

Scenario one is straightforward. The law of large numbers works in your favour. You’ll end up with roughly the expected value.

Scenario two is where things go wrong. A losing streak doesn’t just reduce your balance — it reduces the base you’re betting from. Win 40%, then lose 40%, and you don’t break even. You’re down to 84% of where you started. Each loss hurts proportionally more because you have less to recover with.

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 for £100 million, even though the expected value is over £40 million.⁴ 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.

Some get rich. Most go broke. The median path tells you what typically happens, and it’s rather different from the mean.

The Kelly Criterion

John Kelly, a researcher at Bell Labs, solved this in 1956.⁵ The question isn’t “should I bet?” but “how much?”

For a biased coin at even money,⁶ the formula is:

$f^* = 2p - 1$

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

The intuition matters more than the derivation.⁷ Kelly maximises the expected log growth rate of your wealth. Expected value says bet everything. Kelly says bet enough that the geometric growth rate — the rate at which your money actually compounds over time — is highest.

Kelly wasn’t the first to see this. Daniel⁸ Bernoulli 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 £20,000.⁹ Kelly rediscovered the same mathematics through information theory, two centuries later.

The curve above shows the growth rate for each possible bet fraction. At the Kelly fraction, growth is maximised. Bet more and you enter a regime where losses compound faster than wins — despite positive expected value. Bet 100% and you’re guaranteed to go bust eventually.

At twice Kelly, the expected log growth rate is zero. You’re back to a coin flip in terms of long-term wealth, despite a positive edge. Go beyond that, and you’re expected to lose money — from a favourable bet. Over-betting is worse than not betting at all.

Try again with the Kelly marker visible. The green line on the slider shows where Kelly says you should bet.

The chart above compares full Kelly, half Kelly, and all-in strategies. In practice, many practitioners bet a fraction of Kelly — typically half — trading growth for a smoother ride.¹⁰

Why You Already Knew This

We instinctively resist betting everything, even on good odds. Behavioural economists have spent decades cataloguing this as loss aversion — the finding that losses loom roughly twice as large as equivalent gains.¹¹ The standard interpretation is that we’re irrational; we overweight losses because of some evolutionary hangover.

But think about what compounding actually means. When outcomes compound — and in life, they almost always do — big losses really are worse than they look at face value. Lose 80% of your balance and you haven’t lost 80% of your future; you’ve lost the compounding base that everything else was going to grow from. Going from $10,000 to $2,000 doesn’t just feel catastrophic; in a compounding system, it is catastrophic, because it takes exactly as long to grow $2,000 to $10,000 as it does to grow $10,000 to $50,000. The loss didn’t cost you $8,000. It cost you your next $40,000.¹²

Loss aversion might not be a bias at all. It might be a rough-and-ready heuristic for the geometric expectation — the right calculation, done by feel rather than formula.

Consider laptop insurance. You pay £8 a month to cover a £1,200 machine. The expected claim rate on consumer electronics is something like 15–20% over two years; the insurer charges you £192 over that period for an expected payout of around £200. By expected value, it’s roughly a wash.

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 an 80% drawdown on your available capital. For the insurer, your claim is one of fifty thousand; it barely registers. The same event — one broken laptop — is a rounding error for them and a compounding catastrophe for you. You’re both doing the right calculation. You just have different balances.¹³

This is why you buy phone insurance despite the “bad odds,” why you take the fixed-rate mortgage, why you overpay for certainty in a dozen small ways that expected-value reasoning calls irrational. You’re not failing at arithmetic. You’re doing geometry.

Diversification

What if you could split your bet across several coins? Instead of one coin deciding your fate each round, spread the same wager across several. Each coin is independent — some land heads, some tails — and the variance shrinks.

With enough coins, a single round starts to resemble the ensemble average. You’re turning a sequential bet into something closer to a parallel one. The Kelly fraction stays the same, but the ride gets smoother — and you’re suddenly more willing to bet.

The maths behind this is elegant: variance shrinks as $1/\sqrt{n}$, where $n$ is the number of independent bets.¹⁴ Five coins doesn’t halve your risk, but it cuts it by more than half ($1/\sqrt{5} \approx 0.45$). This is why diversification works, and why Kelly accounts for it.

But diversification has a stranger trick up its sleeve. Consider two coins, each fair, each paying 2:1 on heads and costing you half your stake on tails. Played alone, each coin goes nowhere in the long run — the geometric growth rate is exactly zero ($\sqrt{2 \times 0.5} = 1$). These are not good bets. You’d be right to walk past them.

Now split your money 50/50 across both coins and rebalance after every round. Four outcomes, equally likely:

• Both heads: you double. ($2.00$)
• Both tails: you halve. ($0.50$)
• One heads, one tails: you end up at $1.25$. ($0.5 \times 2 + 0.5 \times 0.5$)

The mixed outcome — one win, one loss — happens half the time, and it leaves you up 25%. The geometric mean across all four outcomes is $(2 \times 1.25 \times 1.25 \times 0.5)^{1/4} \approx 1.12$. That’s 12% growth per round, from two bets that individually grow at 0%.¹⁵

This feels like a magic trick. Where does the growth come from? Rebalancing forces you to sell whichever coin just won and buy whichever just lost — mechanically buying low and selling high, every single round. The growth isn’t hidden in either coin; it lives in the act of rebalancing between them. No individual asset needs to be a winner. The portfolio wins because it harvests the volatility.

The practical upshot: a bet you’d refuse on its own can become worth taking as part of a balanced portfolio. Your gut says “that’s a losing bet, stay away.” The geometry says “that’s a losing bet, but pair it with another losing bet and rebalance.” Intuition gets this one wrong — and it’s worth knowing when it does.

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. 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.