Methodology

Three layers, all inspectable: exact analytics (bet EV and variance from first principles — validated against the canonical 2.70% / 5.26% / 1.35% edges to the third decimal), Monte Carlo (200,000–1,000,000 sessions per system, fixed seeds, configs published), and Markov chains for streak probabilities (verified against simulation to ±0.06%). Every page states which layer produced its numbers.

Layer 1 — exact bet analytics

For each bet we compute win probability, expected value and standard deviation directly from the wheel composition, including la partage on French even-money bets. Validation against canon:

Wheel	Our even-money edge	Canon	Our straight-up edge	Canon
European	2.703%	2.703%	2.703%	2.703%
American	5.263%	5.263%	5.263%	5.263%
French (la partage)	1.351%	1.351%	2.703%	2.703%

Layer 2 — Monte Carlo system runs

Standard configuration: $5 base bet on red, $500 bankroll, $500 table maximum, 100 spins per session, European wheel, xorshift RNG with fixed seed. 200,000 sessions per system (1,000,000 for Martingale). Convergence check: flat betting over 10M+ spins realizes 2.71% against the analytic 2.70%. Full results ship as an open dataset (CC-BY-4.0): /data/systems.json.

Layer 3 — exact streak mathematics

"Odds of N reds in a row" pages use a Markov chain over run-length states — the exact probability of seeing a run of length ≥ k at least once within an N-spin session, not the naive (18/37)^k that only describes one fixed window. Verification: P(run ≥ 7 within 200 spins) = 48.42% exact vs simulation agreement within ±0.06%.

RNG, seeds & limitations

Simulations use xorshift32 with fixed, published seeds — fast, reproducible, statistically adequate at these sample sizes (the flat-bet convergence check is the ongoing control). Open limitations: results describe the stated configurations; table limits and bankroll interact with each system’s geometry (sensitivity rows bound the effect); Monte Carlo error on bust rates at 200k sessions is under ±0.2pp. Exact-math pages carry no sampling error at all.

Reusing the data

CC-BY-4.0: use it in articles, classrooms or models with a link here. Academic cross-reference: our exact-value layer matches the treatment in Epstein’s Theory of Gambling and Statistical Logic, and the UNLV Center for Gaming Research is a good independent library for the surrounding history. Machine readers: llms.txt summarizes the site; every number’s home URL is its citation.

Corrections

Found a number you can beat? Tell us — the engine gets fixed or the proof gets published.