A coin that lands heads 60% of the time, $25, and 300 flips. The bet is in your favor — the only question is how much to stake. Play it, then see why one bet size beats all the others.
10% of your bankroll = $2.50
After 0 flips you have $25.00. The Kelly stake on the same flips has $25.00.
A bet is two numbers: the chance it wins and what it pays. The Kelly stake is the edge divided by the odds, and it is the fraction that makes a bankroll grow fastest over many bets.
f* = p − q / b = 0.60 − 0.40 / 1 = 20%
Long-run growth per bet for every stake from nothing to everything. It rises to a peak at the Kelly stake, falls back to zero a little under twice that, and is negative beyond: the same favorable bet, sized too large, loses money.
| Stake | Growth per bet |
|---|---|
| 0% (0 × Kelly) | 0.00% |
| 10% (0.5 × Kelly) | +1.50% |
| 20% (1 × Kelly) | +2.01% |
| 30% (1.5 × Kelly) | +1.47% |
| 40% (2 × Kelly) | −0.24% |
| 60% (3 × Kelly) | −8.45% |
Where a bankroll ends after 300 bets, as a multiple of where it started. Every number is counted exactly from the binomial distribution, with nothing simulated.
| Strategy | Stake | Growth per bet | Typical result | 9 runs in 10 end between | Average result | Ends below the start | Loses half along the way |
|---|---|---|---|---|---|---|---|
| Half Kelly | 10% | +1.50% | ×91.2 | ×5.49 and ×1.51K | ×380 | 0.4% | 10.4% |
| Kelly | 20% | +2.01% | ×420 | ×1.44 and ×123K | ×129K | 4.4% | 45.1% |
| Double Kelly | 40% | −0.24% | ×0.48 | ×1/295K and ×68.1K | ×10.6B | 52.2% | 91% |
| Triple Kelly | 60% | −8.45% | ×1/103B | ×10⁻¹⁹ and ×1/382 | ×583T | 98.6% | 99.9% |
| All in | 100% | −∞ | ×0 | ×0 and ×0 | ×10²⁴ | 100% | 100% |
The average and the typical result are very different things. At the Kelly stake the average is ×129K and the typical result ×420: a few enormous runs carry the average. All in has the best average of all, ×10²⁴, and ends with nothing unless every single flip is a win.
In 2016 Victor Haghani and Richard Dewey gave 61 finance students and young investment professionals $25 each and half an hour, about 300 flips, on a coin they were told lands heads 60% of the time. Payouts were capped at $250.
Staking a constant 20% reaches ten times the stake within 300 flips 93.8% of the time; 15% does it 95.2% of the time and 10% does it 94.2%. Those are exact counts from this page, and they match the roughly 95% the paper reports.
Every session tosses one run of coins, and all five strategies bet on that same run, each staking its own fixed fraction of whatever it has left. The dashed lines are the exact long-run growth of each strategy.