Kelly Criterion: The Mathematical Formula to Optimize Your Stakes

You've found value. The Lakers at 2.10 when you estimate a 52% probability of winning. Positive EV of +9%.

Now, how much should you bet?

$10? $50? $200? Your entire bankroll?

The Kelly Criterion answers this question with mathematical precision. Developed by John Kelly in 1956, this formula calculates the optimal stake to maximize long-term bankroll growth.

It is the ultimate tool for the quantitative bettor.

What Is the Kelly Criterion?

The Origin

John Kelly Jr., a mathematician at Bell Labs, developed this formula in 1956 to optimize bets on horse racing using insider information.

Since then, it has become the gold standard in: - Sports betting - Financial trading - Portfolio management - Poker

The Principle

The Kelly Criterion determines the optimal percentage of your bankroll to wager on a bet, based on: - Your edge (estimated advantage) - The odds offered

The Basic Formula

Kelly % = (bp - q) / b

Where:
b = decimal odds - 1 (the net profit per dollar wagered)
p = estimated probability of winning
q = probability of losing (1 - p)

Simple Example

You estimate a 55% chance of winning. The odds are 2.00.

b = 2.00 - 1 = 1.00
p = 0.55
q = 0.45

Kelly % = (1.00 × 0.55 - 0.45) / 1.00
Kelly % = (0.55 - 0.45) / 1.00
Kelly % = 0.10 = 10%

Optimal stake: 10% of your bankroll.

The Detailed Formula

Decimal Odds Version

In sports betting, this simplified version is commonly used:

Kelly % = (Odds × Probability - 1) / (Odds - 1)

Or even simpler:
Kelly % = EV% / (Odds - 1)

Reference Table

Probability	Odds 1.50	Odds 1.91	Odds 2.00	Odds 2.50	Odds 3.00
50%	-25%	2.4%	0%	-16.7%	-25%
52%	-22%	4.2%	4%	-12.8%	-22%
55%	-17.5%	7.4%	10%	-6.7%	-17.5%
58%	-13%	10.6%	16%	-0.5%	-13%
60%	-10%	12.6%	20%	4%	-10%
65%	-2.5%	17.9%	30%	14.3%	-2.5%

Note: Negative values = no value, do not bet.

Interpreting the Result

Kelly %	Meaning
Negative	No value — do not bet
0-2%	Slim edge — minimum stake
2-5%	Standard edge — normal stake
5-10%	Strong edge — confident stake
10-20%	Very strong edge — double-check your model
> 20%	Exceptional edge — likely an error

Kelly in Practice

Example 1: Standard Bet

Situation: Chiefs -3 at 1.91, you estimate a 54% probability.

Odds = 1.91
Probability = 0.54

Kelly % = (1.91 × 0.54 - 1) / (1.91 - 1)
Kelly % = (1.0314 - 1) / 0.91
Kelly % = 0.0314 / 0.91
Kelly % = 3.45%

Kelly stake: 3.45% of your bankroll.

On a $1,000 bankroll, that is a $34.50 wager.

Example 2: Underdog with Value

Situation: Browns ML at 3.50, you estimate a 32% probability.

Odds = 3.50
Probability = 0.32

Kelly % = (3.50 × 0.32 - 1) / (3.50 - 1)
Kelly % = (1.12 - 1) / 2.50
Kelly % = 0.12 / 2.50
Kelly % = 4.8%

Kelly stake: 4.8% of your bankroll.

Despite only a 32% win probability, the edge justifies a significant wager.

Example 3: Heavy Favorite with No Value

Situation: Celtics ML at 1.25, you estimate a 75% probability.

Odds = 1.25
Probability = 0.75

Kelly % = (1.25 × 0.75 - 1) / (1.25 - 1)
Kelly % = (0.9375 - 1) / 0.25
Kelly % = -0.0625 / 0.25
Kelly % = -25%

Negative Kelly: DO NOT bet. Even with a 75% win probability, the odds are too short.

The Problem with Full Kelly

Why Full Kelly Is Dangerous

The optimal Kelly Criterion is extremely aggressive. Staking 10% on a single bet sounds reasonable, but:

5 consecutive losses at Kelly 10%:
Remaining bankroll = 0.90^5 = 59%

You have lost 41% of your bankroll.

Full Kelly Variance

Scenario	Probability	Impact
Worst drawdown over 1 year (55% WR)	95%	-50% or more
Bankroll doubled in 1 year	50%	Possible
Bankroll wiped out	~2%	Rare but real

The Estimation Problem

Kelly assumes your probability estimate is exact. In reality:

Your estimate: 55%
Reality: 52%

Calculated Kelly: 10% (based on 55%)
True optimal Kelly: 4% (based on 52%)

You are overbetting by 150%!

Fractional Kelly

The Solution: Scale Down the Kelly

Fractional Kelly reduces the stake to account for estimation uncertainty:

Fraction	Stake	Risk	Growth
Full Kelly (100%)	10%	Very high	Maximum theoretical
Half Kelly (50%)	5%	Moderate	~75% of optimal
Quarter Kelly (25%)	2.5%	Low	~50% of optimal
Eighth Kelly (12.5%)	1.25%	Very low	~25% of optimal

Why Half Kelly Is Recommended

Half Kelly offers the best trade-off:

Full Kelly:
- Maximum theoretical growth
- 50%+ drawdowns are frequent
- Non-negligible risk of ruin

Half Kelly:
- 75% of Full Kelly's growth
- Drawdowns cut in half
- Risk of ruin is virtually zero

Most professional bettors use 1/4 to 1/2 Kelly.

Calculating Fractional Kelly

Stake = Kelly % × Fraction

Example: Kelly 8%, Half Kelly (50%)
Stake = 8% × 0.50 = 4%

Fractional Kelly Table

Calculated Kelly	Full	Half (50%)	Quarter (25%)
2%	2%	1%	0.5%
5%	5%	2.5%	1.25%
8%	8%	4%	2%
10%	10%	5%	2.5%
15%	15%	7.5%	3.75%

Kelly with Multiple Simultaneous Bets

The Concurrency Problem

What if you have 5 simultaneous bets, each at Kelly 5%?

5 × 5% = 25% of your bankroll at risk simultaneously

If all lose: -25% of your bankroll

The Solution: Scaled Kelly

Scale down the allocation based on the number of concurrent bets:

Simultaneous Bets	Total Allocation	Reduction per Bet
1	100% of Kelly	1x
2-3	80% of Kelly	0.8x
4-5	60% of Kelly	0.6x
6-10	50% of Kelly	0.5x
10+	40% of Kelly	0.4x

Practical Example

5 bets today, each with a calculated Kelly of 4%

Reduction: 60%
Stake per bet: 4% × 0.60 = 2.4%

Total exposure: 5 × 2.4% = 12% (vs. 20% without scaling)

Kelly by Bet Type

Spreads and Totals (Odds ~1.91)

For standard-odds bets (-110 / 1.91):

Estimated Probability	Kelly	Half Kelly
52%	2.4%	1.2%
53%	3.5%	1.75%
54%	4.6%	2.3%
55%	5.7%	2.85%
56%	6.8%	3.4%

Favorite Moneylines (Odds < 1.50)

Short-priced favorites require a substantial edge:

Odds	Prob. for Kelly = 0	Prob. for 2% Kelly
1.25	80%	85%
1.40	71.4%	76%
1.50	66.7%	71%

Value on heavy favorites is rare — Kelly is often negative or negligible.

Underdog Moneylines (Odds > 2.50)

Underdogs can produce a high Kelly even with a low win probability:

Odds	Prob. for Kelly = 0	Prob. for 5% Kelly
2.50	40%	47.5%
3.00	33.3%	40%
4.00	25%	31.25%
5.00	20%	26%

Common Kelly Mistakes

Mistake #1: Overestimating Your Edge

Wrong: "I estimate 60%, Kelly says 20%, I bet 20%"
Right: "My estimate may be optimistic, I scale down to 1/4 Kelly = 5%"

Mistake #2: Using Full Kelly

Wrong: "The optimal Kelly is 12%, I bet 12%"
Right: "Kelly says 12%, I bet 3-6% (1/4 to 1/2 Kelly)"

Mistake #3: Ignoring Bet Correlation

Wrong: "3 bets on the same game, 5% each"
Right: "3 correlated bets = treat as 1 single bet, 5% max"

Mistake #4: Applying Kelly with Insufficient Data

Wrong: "My model says 58% on this obscure matchup"
Right: "Insufficient data = high uncertainty = reduce the Kelly"

Mistake #5: Forgetting to Recalculate

Wrong: "I always bet 2% of my bankroll"
Right: "Kelly varies based on the edge and odds of each individual bet"

When Not to Use Kelly

Scenario 1: Uncertain Edge

If you are not confident in your probability estimate:

"Confident" estimate: Standard Kelly (1/2)
"Uncertain" estimate: Reduced Kelly (1/4)
"Speculative" estimate: Fixed minimum stake

Scenario 2: Small Sample Size

Your model is based on limited data:

100+ historical games: Standard Kelly
50-100 games: Reduced Kelly (1/2)
< 50 games: Minimum stake or pass

Scenario 3: Illiquid Market

Player props, minor leagues, exotic markets:

NFL main market: Standard Kelly
Player props: Reduced Kelly (1/4)
Minor league: Fixed minimum stake

Scenario 4: Implausibly High Edge

A Kelly above 15% almost certainly indicates an error:

Calculated Kelly > 15%?
-> Verify your probability estimate
-> Verify the odds (book pricing error?)
-> If everything checks out: cap at 5% anyway

Integrating Kelly into Your Overall Strategy

Recommended Approach for Beginners

1. Use flat betting (fixed stake) for 3-6 months
2. Build a track record to measure your actual edge
3. Move to Fractional Kelly (1/4) once your edge is validated
4. Gradually increase (1/3, 1/2) based on results

Approach for Experienced Bettors

1. Calculate Kelly for every bet
2. Apply 1/4 to 1/2 Kelly based on confidence
3. Scale down for simultaneous bets
4. Cap at 5% maximum per bet
5. Track CLV to validate your edge

The Tiered Kelly System

Combine Kelly with confidence tiers:

Tier	Confidence in the Edge	Kelly Applied	Max Stake
Standard	Model suggests value	1/4 Kelly	2%
Confident	Clear edge, solid data	1/2 Kelly	3%
High conviction	Major edge, optimal timing	3/4 Kelly	5%

Full Example: A Week of Kelly Betting

Your Bankroll

Bankroll: $2,000
Strategy: Quarter Kelly (1/4)
Cap per bet: 5% ($100)

The Week's Bets

Day	Bet	Est. Prob.	Odds	Kelly	1/4 Kelly	Stake
Mon	Chiefs -3	54%	1.91	4.6%	1.15%	$23
Tue	Lakers ML	48%	2.15	3.5%	0.88%	$18
Wed	Under 47 NFL	55%	1.95	6.8%	1.7%	$34
Thu	Celtics -5.5	56%	1.91	6.8%	1.7%	$34
Fri	Browns +7	42%	1.91	3.5%	0.88%	$18
Sat	Over 8.5 NHL	57%	1.87	5.6%	1.4%	$28
Sun	Ravens ML	62%	1.75	8.7%	2.2%	$44

The Bottom Line

Total wagered: $199
If 4/7 winners (57%): profit ~+$40
Max simultaneous exposure: ~3-4% (very manageable)

How ProbWin Uses the Kelly Criterion

Our Approach

At ProbWin, we calculate Kelly for every pick but recommend a conservative approach:

Our process:
1. Model calculates the estimated probability
2. Kelly is computed based on market odds
3. Quarter Kelly (1/4) is applied systematically
4. Standard picks are capped at 3%
5. Up to 5% for "high confidence" picks

Why We Recommend Quarter Kelly

1. Estimation uncertainty is real
2. Variance is significantly reduced
3. Growth remains solid (50% of optimal)
4. Risk of ruin is virtually zero
5. It is far more psychologically sustainable

Our Pick Categories

Category	Typical Kelly	Our Stake	Frequency
Standard	3-5%	1-1.5%	70%
Confident	5-8%	1.5-2%	25%
High conviction	8%+	2-3%	5%

Check out our picks with confidence levels included.

Tools to Calculate Kelly

Simple Calculator

Kelly % = (Odds × Probability - 1) / (Odds - 1)

Quick example:
Odds 1.91, Prob 55%
Kelly = (1.91 × 0.55 - 1) / 0.91 = 5.7%

Kelly Spreadsheet

Build a spreadsheet:

Column	Formula
A	Decimal odds
B	Estimated probability (%)
C	=(A1*B1/100-1)/(A1-1) (Kelly %)
D	=C1*0.25 (Quarter Kelly)
E	=D1*BankrollTotal (Stake $)

Apps and Websites

Tool	Type	Cost
Kelly Calculator (web)	Free	$0
Bet Tracker apps	Mobile	$0-5
Excel/Sheets	DIY	$0

Summary: The Kelly Criterion in 7 Points

#	Key Takeaway
1	Kelly % = (Odds x Prob - 1) / (Odds - 1)
2	Full Kelly is too aggressive — use 1/4 to 1/2 Kelly
3	Negative Kelly = no value, do not bet
4	Reduce Kelly when the edge is uncertain or data is limited
5	Scale down the allocation for simultaneous bets
6	Kelly > 15% = probably an error, double-check
7	Always cap at 5% max per bet

Next Step

You now know how to calculate optimal bet sizing. For NFL bettors, there is a specific strategy that exploits Key Numbers: Teasers.

Learn about NFL Teasers — how to buy points intelligently to cross key numbers and improve your win probability.