Methodology & Formulas
Every calculator on this site is built on a published, verifiable formula. Here's the math, the sources, and the verification process.
// Principles
Transparency
Every formula we use is published below with its source. No hidden constants, no proprietary 'secret sauce' β if you understand high-school algebra, you can verify our outputs by hand.
Verification
Each calculator is tested against reference values from academic literature, regulator publications, or operator official paytables. Discrepancies are caught before release.
Peer review
Math-heavy articles are reviewed by a second domain expert before publishing. Where peer-reviewed academic results exist (Kelly, Thorp, Shackleford), we cite the original paper, not a downstream blog post.
Limitations stated
Every calculator includes the assumptions that make its output valid. Kelly assumes a known edge. RTP assumes long-run play. Arbitrage assumes simultaneous bet acceptance. We never hide the constraints behind a formula.
// Core formulas
These six formulas underpin almost every calculator on the site. Each is sourced from a published, peer-reviewed origin.
Kelly Criterion β optimal bet sizing
f* = (bp β q) / bKelly returns the bet fraction that maximizes long-run logarithmic growth of bankroll. b = decimal odds minus 1. p = win probability. q = 1 β p. We default to fractional Kelly (ΒΌ Kelly) in calculator outputs because full Kelly assumes perfect knowledge of edge and tolerates ~30% drawdowns.
SOURCES: Kelly J.L. (1956) β A New Interpretation of Information Rate, Bell System Technical Journal. Thorp E.O. (1969) β Optimal gambling systems for favorable games.
Expected Value β every bet's true cost
EV = (P_win Γ Payout) β (P_lose Γ Stake)Expected value is the average outcome of a bet repeated infinitely. Positive EV means the bet is profitable in the long run; negative EV means it isn't. Every odds-converter, parlay calculator, and bonus-EV tool on this site computes EV before showing a recommendation.
SOURCES: Standard probability theory. Reference: Ross S. (2010) β A First Course in Probability.
Return to Player β slot/table game payout rate
RTP = Ξ£(Payout Γ Probability) Γ 100%RTP is the long-run percentage of wagers a game returns to players. Calculated as the sum of (payout Γ probability) across all outcomes. RTP only converges over millions of spins β short-term variance dominates real sessions.
SOURCES: GLI-19 testing standard. eCOGRA monthly RTP reports. NJDGE published certification data.
House Edge β built-in casino margin
House Edge = 100% β RTPThe casino's mathematical advantage. Defined as 100% minus RTP. A 96% RTP slot has a 4% house edge β over a long session, the house keeps 4% of every dollar wagered. Lower edge = better game for the player.
SOURCES: Shackleford M. (Wizard of Odds) β published edge tables for table games. Operator-published paytables.
Arbitrage β bookmaker price discrepancy
1/odds_A + 1/odds_B < 1 β arbitrage existsIf the sum of inverse decimal odds across all outcomes is below 1, an arbitrage exists β a guaranteed profit regardless of result, by staking proportionally across bookmakers. Real arbitrages are short-lived (under 30 seconds at sharp books) and constrained by max-stake limits.
SOURCES: Standard derivative pricing math. Reference: Hull J. (2018) β Options, Futures, and Other Derivatives.
Wagering β true cost of a bonus
Effective Cost = Bonus Γ Wagering Γ House EdgeA bonus's apparent value rarely equals its real value. After applying wagering requirements (e.g. 30Γ bonus + deposit) and the house edge of the games that count toward wagering, the expected value to the player is typically 20-40% of the bonus face value, not 100%.
SOURCES: UKGC bonus-disclosure rules. eCOGRA wagering compliance reports. Operator-published bonus terms.
// Verification process
Before any calculator is published:
- The formula is reproduced from a primary source (academic paper, regulator filing, or operator paytable).
- Test cases are constructed from the source's worked examples. The calculator must match to four decimal places.
- Edge cases β zero stakes, infinite bankrolls, certainty bets β are tested explicitly.
- Output is compared against at least one independent online calculator from a reputable source (e.g. Wizard of Odds, an academic reference implementation).
- The tool ships only after passing all four steps.
Calculators are re-verified whenever the underlying formula source publishes a correction, or when a user reports a discrepancy.
// What our calculators don't do
Honesty about limitations is part of the methodology:
- We don't predict random outcomes. RNG is RNG. Probability calculators tell you the math, not the next spin.
- We don't model correlated bets perfectly. Same-game parlays have correlation our default formulas may not capture; we flag this where relevant.
- We don't replace operator rules. Always read the actual T&Cs of the operator you're using. Our wagering math assumes their stated terms are accurate; if they aren't, our output will be too.
- We don't account for variance over short sessions. A 96% RTP slot will not return $96 of every $100 in any single session. Over a million spins, it will. Use bankroll calculators for short-run risk.