Formula Reference

12 essential formulas for quantitative prop betting

Pricing

Convert odds, calculate break-even rates, and quantify expected value.

Implied Probability

Ch. 2

Convert American odds to the probability embedded in the price. The foundation of all quantitative betting analysis.

If odds < 0: IP = |odds| / (|odds| + 100)
If odds > 0: IP = 100 / (odds + 100)

Variables

oddsAmerican odds (e.g., -110 or +150)

IPImplied probability (0 to 1)

Example

-110 → 110 / (110 + 100) = 52.38% | +150 → 100 / (150 + 100) = 40.00%

Excel=IF(A1<0, ABS(A1)/(ABS(A1)+100), 100/(A1+100))

Break-Even Win Rate

Ch. 2

The percentage of wins required to produce an EV of exactly $0 at a given price. Any true win rate above this threshold represents +EV.

BE% = Implied Probability (with vig included)

Variables

BE%Break-even win percentage

oddsThe price at which you are betting

Example

At -110: BE% = 110 / 210 = 52.38%. You must win more than 52.38% to profit.

Excel=IF(A1<0, ABS(A1)/(ABS(A1)+100), 100/(A1+100))

Expected Value (EV)

Ch. 4

The average expected profit or loss per wager over an infinite number of repetitions. The core metric for evaluating any bet.

EV = (p × profit) − ((1 − p) × stake)

Variables

pYour estimated true win probability

profitAmount won if the bet wins

stakeAmount risked

Example

p = 55%, odds = -110, stake = $100: EV = (0.55 × $90.91) − (0.45 × $100) = +$5.00

Excel=(B1*C1)-((1-B1)*D1)

Bankroll

Optimal position sizing and performance measurement.

Kelly Criterion (Fractional)

Ch. 5

The optimal position-sizing formula to maximize the geometric growth rate of your bankroll while controlling risk of ruin.

f* = (bp − q) / b

Variables

f*Fraction of bankroll to wager

bDecimal odds profit per $1 risked (win/risk)

pEstimated true win probability

q1 − p (loss probability)

Example

p = 55%, odds = -110: b = 0.909, f* = (0.909 × 0.55 − 0.45) / 0.909 = 5.5% of bankroll

Excel=((B1*C1)-(1-C1))/B1

ROI (Return on Investment)

Ch. 4

Measure total return as a percentage of capital deployed. The benchmark metric for comparing performance across bettors and strategies.

ROI = (Net Profit / Total Risked) × 100

Variables

Net ProfitTotal winnings minus total losses

Total RiskedSum of all amounts wagered

Example

$500 profit on $10,000 risked = 5.0% ROI

Excel=(SUM(winnings)-SUM(stakes))/SUM(stakes)*100

Distributions

Statistical models for pricing props and measuring spread.

Poisson Probability (PMF)

Ch. 8

Calculate the exact probability of observing k events given an expected rate λ. Used for pricing "counting" props like strikeouts, touchdowns, and goals.

P(X = k) = (λ^k × e^(-λ)) / k!

Variables

kNumber of events (e.g., 2 touchdowns)

λExpected count (e.g., 0.8 TDs/game)

eEuler's number (~2.71828)

k!Factorial of k

Example

λ = 0.8, k = 1: P(1 TD) = (0.8¹ × e⁻⁰·⁸) / 1! = 35.9%

Excel=POISSON.DIST(A1, B1, FALSE)

Normal Distribution (CDF)

Ch. 7

Calculate the probability that a normally distributed variable falls below a given threshold. Used for continuous props like passing yards and fantasy points.

P(X ≤ x) = Φ((x − μ) / σ)

Variables

xThreshold value (e.g., 249.5 passing yards)

μMean (average performance)

σStandard deviation

ΦStandard normal CDF

Example

μ = 245, σ = 55, line = 249.5: Z = 0.082, P(Under) = 53.3%

Excel=NORM.DIST(A1, B1, C1, TRUE)

Standard Deviation

Ch. 6

Measure the "spread" of player performance around their mean. Critical for setting confidence intervals and identifying outlier performances.

σ = √(Σ(xᵢ − μ)² / n)

Variables

σStandard deviation

xᵢEach individual observation

μPopulation mean

nNumber of observations

Example

A QB with mean 245 yds and σ = 55 yds: ~68% of games fall between 190–300 yds

Excel=STDEV.S(A1:A20)

Z-Score

Ch. 7

Convert any value to standardized units for probability lookup. The bridge between raw data and the normal distribution probability table.

Z = (X − μ) / σ

Variables

XObserved or test value

μMean

σStandard deviation

ZNumber of standard deviations from the mean

Example

Passing yards: X = 280, μ = 245, σ = 55: Z = (280 − 245) / 55 = +0.636

Excel=(A1-B1)/C1

VMR (Variance-to-Mean Ratio)

Ch. 9

Diagnostic test to determine whether count data is overdispersed. Guides the choice between Poisson (VMR ≈ 1) and Negative Binomial (VMR > 1) models.

VMR = σ² / μ

Variables

σ²Sample variance

μSample mean

VMRRatio (1.0 = equidispersed)

Example

Mean = 4.5, Variance = 8.1: VMR = 8.1 / 4.5 = 1.80 → Overdispersed, use Negative Binomial

Excel=VAR.S(A1:A20)/AVERAGE(A1:A20)

Advanced

Edge quantification and multi-leg correlation analysis.

CLV (Closing Line Value)

Ch. 12

Quantify your edge by comparing your entry price to the final efficient market price. The most reliable long-term predictor of betting profitability.

CLV = IP_close − IP_entry

Variables

IP_closeImplied probability at market close

IP_entryImplied probability at your entry

Example

Bet at -110 (52.38%), closed at -125 (55.56%): CLV = +3.18%

Excel=IF(B1<0,ABS(B1)/(ABS(B1)+100),100/(B1+100))-IF(A1<0,ABS(A1)/(ABS(A1)+100),100/(A1+100))

Correlation Coefficient

Ch. 11

Measure the strength and direction of the linear relationship between two prop outcomes. Essential for correctly pricing same-game parlays.

r = Cov(X, Y) / (σₓ × σᵧ)

Variables

Cov(X,Y)Covariance between variables X and Y

σₓStandard deviation of X

σᵧStandard deviation of Y

rCorrelation coefficient (−1 to +1)

Example

Passing yards and passing TDs: r ≈ 0.45 → Moderate positive correlation

Excel=CORREL(A1:A20, B1:B20)