Small edge. Small footsteps.
Slow, imperfect progress. The bankroll remains recognizable throughout the trip.
A GAME YOU ARE SUPPOSED TO WIN
You can be
right and still
go broke.
The coin is favorable. The odds are known. The only decision left is how much of your money to risk.
Here is the entire game. A coin lands heads 55% of the time. Heads wins whatever you bet. Tails loses it. You have $10,000 and one hundred chances to play.
No fees. No changing odds. No hidden trick.
The edge is already decided.
You decide the danger.
FAVORABLE COIN CONNECTED
HEADS 55 · TAILS 45
BET SIZE
of whatever money remains
1% cautious50% violent
FIRST BET$2,500
BANKROLL$10,000
FLIP0 / 100
RECORD0–0
?
YOU WON 55 OF 100 BETS
—
The same edge can create wealth or ruin. The bet size chooses which experience is available.
NOT A BAD STREAK
You were right more often than you were wrong.
The coin behaved exactly as promised. Fifty-five wins. Forty-five losses. If your bankroll suffered, bad luck is not the only suspect.
Positive expectancy can tell you to play.
It cannot tell you how much to bet.
THE MULTIPLICATION TRAP
One win. One loss.
Back where you started?
START$100
×
WIN 40%1.40
×
LOSE 40%0.60
=
FINISH$84
A percentage gain and an equal percentage loss do not cancel. They multiply. Every deep loss shrinks the base available to recover.
SAME COIN · SAME ORDER · FOUR BETTORS
Only one dial moves.
Scroll through four people betting on the exact one hundred flips you just saw. Nothing changes except the fraction of wealth they risk.
The mathematical summit.
In this perfectly known game, ten percent maximizes the expected long-run logarithmic growth rate.
More conviction. Less wealth.
The same 55 wins no longer compensate for the damage inflicted by 45 proportional losses.
Right 55 times. Almost erased.
The edge survived. The bettor did not. After the same favorable sequence, $10,000 becomes roughly $113.
NOW FILL THE CASINO
One bettor is a story.
Ten thousand are a distribution.
Give 10,000 people the same 55% coin for 200 bets. Each receives a different sequence. Choose the fraction they all risk.
10% KELLY
TYPICAL FINISH—
MODEL EXPECTATION—
FINISH BELOW $10,000—
LOSE 90%+—
THE BILLIONAIRE IN THE AVERAGE
The mean can get rich while most people get poor.
Large fractions assign a tiny probability to astronomical winners. Those rare outcomes pull the model’s arithmetic expectation upward—even while the median bettor collapses. Expected dollars and typical compounded experience are answering different questions.
ENTER JOHN KELLY
There is a summit.
For repeated even-money bets, the long-run growth score is the expected logarithm of the wealth multiplier. It rewards compounding and punishes destructive losses.
EVEN-MONEY KELLYf* = p − q
55% − 45% = 10%
Ten percent is not the fraction with the highest possible ending. It is the fraction that maximizes the expected long-run geometric growth rate inside this artificial world.
At roughly —, expected log growth crosses below zero. Beyond that point, the typical compounded path decays even though every individual bet still has positive expected profit.
THE SUMMIT IS NOT COMFORTABLE
Optimal growth can feel untradeable.
Full Kelly maximizes an asymptotic growth objective. It does not minimize drawdown, protect your sleep, or care when you need the money.
FULL KELLY · 200 BETS—
of simulated lives suffer a peak-to-trough loss greater than 50%.
FULL KELLY—
median maximum drawdown across 10,000 lives
THE EDGE IS NEVER PRINTED ON THE COIN
Kelly knows exactly what you do not.
Our game handed you the true 55% probability. A strategy gives you an estimate—built from finite, noisy, possibly selected history.
YOUR MODEL SAYS55%implied full Kelly: 10%
The formula is precise.
Your inputs are not.
FROM FRACTION TO CONTRACTS
A percentage is not yet an order.
Trading requires another translation: account risk divided by the dollars at risk per contract. This calculator is arithmetic—not a recommendation or a Kelly estimate.
ACCOUNT RISK$500
÷RISK / CONTRACT$500
=WHOLE CONTRACTS1
This ignores portfolio overlap, gaps, slippage, margin constraints, changing volatility, and estimation error.
YOUR BET, RECONSTRUCTED
You chose
—.
SAME 55–45 STORY—ending bankroll
10,000-LIFE MEDIAN—after 200 bets
FINISH BELOW START—simulated frequency
An edge tells you which side to bet.
Sizing decides whether you survive being right.
In markets, the odds are uncertain, payoffs change, bets overlap, and the future can leave the historical distribution entirely. That does not make sizing less important. It makes false precision more dangerous.
Experiment, assumptions, and sources
Teaching experiment
Every bet is independent, wins with known probability 0.55, and pays even money. A fraction f produces a wealth multiplier of 1+f after a win and 1−f after a loss. This is synthetic by design.
Story sequence
The opening contains exactly 55 wins and 45 losses in one deterministic shuffled order. Every position size is applied to that identical sequence.
Simulation
10,000 deterministic-seed lives of 200 bets. “Typical” means the median. Drawdown includes the $10,000 starting bankroll as an eligible peak.
Kelly objective
For this even-money binary game, full Kelly is f* = p − q = 10%. It maximizes expected logarithmic growth under the model—not comfort, finite-horizon utility, or certainty.
Trading boundary
Real strategy returns are not known IID coin flips. Probabilities and payoffs are estimated, may change, and can reflect selection bias. The contract calculator is implementation arithmetic only.
Primary sources
Kelly (1956) · Thorp, Optimal Gambling Systems · Thorp on Kelly in gambling and markets