Gambling Online 101
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12 min readKelly Criterion: Model-Based Bet Sizing
A model-based framework for sizing bets when you can estimate edge and risk.
BonusBell Team
You've found a +EV bet. Now what? Bet too little and you leave growth on the table. Bet too much and a bad streak wipes you out. The Kelly Criterion is a mathematical framework for this dilemma—a formula that maximizes long-term log growth when your probability and payout assumptions are correct.
The Problem Kelly Solves
Imagine you have a coin that lands heads 60% of the time, paying even money. You clearly have an edge. But how much of your bankroll should you wager on each flip?
- Bet 1% — Safe, but your bankroll barely grows
- Bet 50% — Aggressive, but a few losses in a row devastate you
- Bet 100% — One loss and you're done forever
Kelly tells you the exact fraction that maximizes the geometricgrowth rate of your wealth. Not the arithmetic average—the geometric rate, which accounts for compounding and the asymmetry between gains and losses.
The Derivation: Maximizing E[log(W)]
The key insight is that we want to maximize E[log(W)]—the expected value of the logarithm of wealth. Why logarithm? Because wealth compounds multiplicatively, not additively. Doubling and then halving leaves you at the same place, not up by the average gain.
For a bet with win probability p, loss probability q = 1 − p, and net odds b (you win b units for every 1 unit risked), the expected log-growth when betting fraction f of your bankroll is:
Log-Growth Function
G(f) = p × ln(1 + f × b) + q × ln(1 − f)=Maximize G(f) to find optimal f
Taking the derivative, setting it to zero, and solving for f gives us the Kelly fraction.
Setting dG/df = 0 and solving:
The Kelly Formula
f* = (b × p − q) ÷ b=f* = optimal fraction of bankroll to wager
Where p = win probability, q = 1 − p (loss probability), and b = net odds (decimal odds − 1). This is equivalent to f* = (edge) ÷ (odds).
Good to Know
An equivalent form that many bettors find more intuitive: f* = edge ÷ odds, where edge = (b × p − q) and odds = b. Your edge is the amount you expect to win per dollar wagered, and you divide by the odds to normalize.
Key Properties of Kelly
- Maximizes geometric growth — No other fixed-fraction strategy grows your bankroll faster in the long run
- Never bets the entire bankroll — Since f* < 1 always, you can never go to zero on a single bet
- Says "don't bet" on −EV wagers — When edge ≤ 0, Kelly returns f* ≤ 0
- Scales with edge — Bigger edge = bigger bet, proportionally
- Self-correcting — As your bankroll grows, bets grow; as it shrinks, bets shrink
Worked Example 1: Standard −110 Bet
You estimate a 55% win probability on a −110 line (decimal odds 1.909, so b = 0.909).
Kelly at −110 with 55% Edge
f* = (0.909 × 0.55 − 0.45) ÷ 0.909 = (0.500 − 0.450) ÷ 0.909=f* ≈ 5.5% of bankroll
With a $1,000 bankroll, Kelly says wager $55. This is a relatively modest edge, so Kelly recommends a modest bet.
Worked Example 2: +200 Longshot
You believe a +200 underdog (decimal odds 3.0, b = 2.0) has a 40% true probability to win.
Kelly on a +200 Longshot
f* = (2.0 × 0.40 − 0.60) ÷ 2.0 = (0.80 − 0.60) ÷ 2.0=f* = 10% of bankroll
The edge is large (expected value = 0.40 × 3.0 − 1 = +20%), so Kelly recommends a substantial 10% of your bankroll. But full Kelly here is extremely aggressive in practice.
Strategy Insight
Notice that the +200 longshot gets a largerKelly fraction (10%) than the −110 bet (5.5%), even though the longshot loses more often. Kelly cares about edge relative to odds, not win frequency alone.
Why Full Kelly Is Too Aggressive
Full Kelly maximizes theoretical log growth under perfect inputs, but in practice it produces stomach-churning drawdowns:
- The median drawdown from peak during full Kelly betting is roughly 50%
- You'll experience a 90% drawdown at some point with meaningful probability
- Kelly assumes you know the exact probabilities—any overestimation of your edge leads to over-betting, which is far more damaging than under-betting
Warning
Over-betting by even a small margin is catastrophic. Betting 2× Kelly produces zero long-term growth. Betting more than 2× Kelly causes your bankroll to converge to zero. Over-betting is far worse than under-betting.
Fractional Kelly: The Practical Solution
Nearly all professional gamblers and institutional investors use fractional Kelly—typically betting a fraction of what full Kelly recommends.
Fractional Kelly Comparison
| Strategy | Growth Rate | Max Drawdown | Best For |
|---|---|---|---|
| Full Kelly (100%) | 100% of max | ~50% median | Theoretical maximum only |
| Half Kelly (50%) | ~75% of max | ~25% median | Most professionals use this |
| Quarter Kelly (25%) | ~44% of max | ~12% median | Conservative / uncertain edge |
| Tenth Kelly (10%) | ~19% of max | ~5% median | Very high variance bets |
The key insight: half Kelly gives you 75% of the maximum growth rate while cutting drawdowns roughly in half. This is an extraordinary risk-reward tradeoff, which is why half Kelly is the gold standard among professionals.
Strategy Insight
Use half Kelly as your default. Drop to quarter Kelly when you're less confident in your edge estimate. The cost of under-betting is linear; the cost of over-betting is exponential. When in doubt, bet smaller.
Simultaneous Kelly for Multiple Bets
When you have multiple simultaneous +EV bets (e.g., several games on the same day), the independent Kelly fractions don't simply add up. The total amount wagered must be constrained to prevent over-exposure.
For independent bets, a practical approach:
- Calculate individual Kelly fractions for each bet
- If the sum exceeds your comfort level (e.g., 20–30% of bankroll), scale them all down proportionally
- For correlated bets, reduce further—correlated losses compound the damage
Scaling Multiple Bets
If 5 bets each have f* = 8%, total = 40%. Scale by (20% ÷ 40%) = 0.5=Each bet becomes 4% of bankroll (20% total exposure)
This proportional scaling preserves the relative sizing between bets (better edge = bigger bet) while capping total risk.
Common Kelly Mistakes
Overestimating Your Edge
If your true win rate is 53% but you think it's 58%, you're over-betting by roughly 2×. This is why fractional Kelly is essential—it provides a buffer against estimation error.
Ignoring Correlation
Betting on multiple outcomes in the same game? Those bets are correlated. Treating them as independent overestimates diversification and leads to over-betting.
Applying Kelly to −EV Bets
Kelly only works when you have a genuine edge. No amount of clever bet-sizing turns a −EV bet into a winning proposition. Step one is always finding +EV opportunities.
Kelly and Bankroll Management
Kelly naturally integrates with bankroll management:
- Your "bankroll" is the amount you can afford to lose entirely—never your rent money
- Recalculate after every bet (your bankroll changes, so bet size should too)
- In practice, update bet sizes daily or weekly rather than after every single wager
Try It: Kelly Criterion Calculator
Edge: 10.0%
| Strategy | % of Roll | Bet Size | Growth |
|---|---|---|---|
| Full Kelly | 10.0% | $100 | 0.50% |
| Half Kelly ★ | 5.0% | $50 | 0.38% |
| Quarter Kelly | 2.5% | $25 | — |
Half Kelly (★) is recommended — 75% of the growth rate with roughly half the drawdown risk.
Good to Know
Size Your Bets Precisely
Use our Kelly Criterion Calculator for full Kelly, half Kelly, and quarter Kelly sizing with growth-rate projections. Pair it with the Risk of Ruin Calculator to understand your probability of going bust at different bet sizes.
Sources & References
- Kelly, J. L. (1956). "A New Interpretation of Information Rate." Bell System Technical Journal, 35(4), 917–926. The original paper deriving the Kelly criterion from information theory.
- Thorp, E. O. (2006). "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." Handbook of Asset and Liability Management, Volume 1. Comprehensive treatment of practical Kelly applications in gambling and finance.
- MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (2011). "The Kelly Capital Growth Investment Criterion." World Scientific. Definitive academic collection covering fractional Kelly, simultaneous bets, and estimation error.
- Growth-rate and drawdown comparisons for fractional Kelly strategies are standard results derivable from the Kelly log-wealth framework. Half Kelly achieving ~75% growth at ~50% drawdown reduction is independently verifiable.
Mathematical claims are independently verifiable. BonusBell platform analysis reflects our tracked platform directory and dated evidence checks as of March 2026.
Key Takeaways
- 1Kelly maximizes long-term bankroll growth by betting f* = (bp − q) ÷ b of your bankroll
- 2Full Kelly is too aggressive in practice—use half Kelly for 75% growth with ~50% less drawdown
- 3Over-betting is far more dangerous than under-betting—at 2× Kelly, growth drops to zero
- 4Kelly only works on +EV bets—no sizing strategy can overcome a negative edge
- 5For multiple simultaneous bets, scale individual Kelly fractions down to cap total exposure