#probability

Lottery combinatorics as an API, computed locally and deterministically and exactly — the real odds behind a ticket, the maths the jackpot poster never shows. The odds endpoint gives the jackpot odds of a pick-N game as the number of possible tickets, C(pool, picks), times the bonus-ball pool if there is one: a 6/49 game is 1 in 13,983,816, a 5/69-plus-1/26 Powerball-style game is 1 in 292,201,338, and every single line is equally unlikely. The match-odds endpoint gives the chance of matching exactly k of the main numbers — a prize tier — from the hypergeometric formula C(picks, k)·C(pool−picks, picks−k) ÷ C(pool, picks), so matching 3 of 6 in a 6/49 game is about 1 in 57. The expected-value endpoint turns a jackpot and ticket price into the expected value and the break-even jackpot (price × the odds), the threshold a jackpot must clear before a ticket is even theoretically worth it — before a shared jackpot, lump-sum and tax pull it back under. Everything is computed locally and deterministically, so it is instant and exact. Ideal for lottery and odds apps, gambling-education and responsible-play tools, probability teaching, and game back-ends. Pure local computation — no key, no third-party service, instant. Exact combinatorics. Live, nothing stored. 3 compute endpoints. Educational — not gambling advice; the odds are always against you.

api.oanor.com/lottery-api

Roulette-Odds-Mathematik als API, lokal und deterministisch und exakt berechnet – die Auszahlung, die wahre Wahrscheinlichkeit und der Hausvorteil hinter jeder Wette, die Zahlen, die Ihnen ein faires Spiel verrät und die ein Casino lieber ignorieren würde. Der Auszahlungs-Endpoint gibt die Auszahlung einer Wette, die Gewinnzahlen, die Gewinnwahrscheinlichkeit und den Hausvorteil für ein europäisches (Single-Zero) oder amerikanisches (Double-Zero) Rad aus: Eine einfache Zahl zahlt 35 zu 1, gewinnt aber nur 1 von 37, ein Vorteil von 2,70 % europäisch oder 5,26 % amerikanisch, bei fast jeder Wette gleich, weil die Auszahlung die Nullen einfach ignoriert. Der Expected-Value-Endpoint wandelt einen Einsatz in seinen Erwartungswert um – Einsatz × (Gewinnwahrscheinlichkeit × (Auszahlung + 1) − 1), immer negativ und gleich minus Einsatz mal Hausvorteil – also €10 auf eine einzelne Zahl auf einem europäischen Rad sind −€0,27 pro Drehung wert. Der Martingale-Endpoint legt das Verdopplungssystem offen: total riskiert = Basis × (2^Schritte − 1), der Einsatz, der nach einer Verlustserie explodiert, und die Pleite-Wahrscheinlichkeit – Beweis der Mathematik, dass keine Progression die Null schlägt. Alles wird lokal und deterministisch berechnet, also ist es sofort und exakt. Ideal für Casino-Spiel- und Odds-Apps, Glücksspiel-Bildungs- und verantwortungsvolle Spiel-Tools, Game-Design-Backends und Wahrscheinlichkeitslehre. Reine lokale Berechnung – kein Key, kein Drittanbieter-Service, sofort. Live, nichts gespeichert. 3 Compute-Endpoints. Lehrreich – kein Wettberatung; das Haus gewinnt langfristig immer.

api.oanor.com/roulette-api

Tabletop dice-probability maths as an API, computed locally and deterministically and exactly — the odds behind the rolls, not the rolls themselves. The advantage endpoint gives the D&D-style chances of beating a target on a d20 (or any die) rolling normally, with advantage (roll twice, keep the higher) or with disadvantage (keep the lower): needing an 11+ is 50 % normally, 75 % with advantage and 25 % with disadvantage, and it reports the average roll — advantage lifts a d20 from 10.5 to about 13.8. The pool endpoint handles success-counting systems (World of Darkness, Shadowrun): for a pool of dice that succeed on a face at or above a threshold it gives the chance per die, the expected number of successes and the exact binomial probability of getting exactly, or at least, a target number — six d10s succeeding on 7+ average 2.4 successes with a 45.6 % chance of three or more. The exploding endpoint gives the mean of an exploding ("acing", open-ended) die that re-rolls and adds on its maximum face — a d6 averages 4.2 instead of 3.5. Everything is computed locally and deterministically, so it is instant and private. Ideal for tabletop, virtual-tabletop, game-design and TTRPG app developers, odds-and-probability helpers, and game-master tools. Pure local computation — no key, no third-party service, instant. Exact maths, no simulation. Live, nothing stored. 3 compute endpoints. For random rolls use a dice-roller API.

api.oanor.com/dicepool-api

Birthday-paradox and collision-probability maths as an API, computed locally and deterministically. The probability endpoint computes the chance that at least two of n people share a birthday among d equally likely days, P = 1 − Π(1 − i/d), evaluated in log space for accuracy — the famous result that just 23 people give about a 50.7 % chance, 50 people about 97 % and 70 people about 99.9 %. The people-needed endpoint inverts it: the smallest group size to reach a target probability (23 for 50 %, 57 for 99 %), with the √(2·d·ln(1/(1−p))) approximation. The collision endpoint generalises the birthday bound to any space — pass a number of buckets or a hash size in bits — and returns the collision probability P ≈ 1 − e^(−n²/2d), the rule behind hash collisions and UUID-uniqueness estimates, where a 50 % chance needs roughly 1.177·√d items. Days and buckets default to 365. Everything is computed locally and deterministically, so it is instant and private. Ideal for probability-education, security, cryptography, hashing, data-engineering and statistics app developers, collision-risk and birthday-problem tools, and teaching material. Pure local computation — no key, no third-party service, instant. Live, nothing stored. 3 endpoints. This is the birthday/collision probability; for full distributions use a probability API.

api.oanor.com/birthdayparadox-api

Betting-odds maths as an API, computed locally and deterministically. The convert endpoint translates a price between every format used by bookmakers — decimal (European), fractional (UK), American (moneyline) and the implied probability — give it any one and it returns all the others, with the implied probability that the odds represent (1 ÷ decimal). The payout endpoint computes the profit and total return for a stake at given decimal or American odds. The parlay endpoint combines several decimal-odds selections into one accumulator by multiplying them, returning the combined odds, the implied probability and the payout for a stake — every leg must win, so the payout grows fast while the probability shrinks. Decimal odds are the total return per unit staked, American odds are at least +100 for an underdog or −100 or lower for a favourite, and fractional odds look like 5/2. Everything is computed locally and deterministically, so it is instant and private. Ideal for sports-betting, fantasy, odds-comparison and gaming app developers, bet-slip and value tools, and probability education. Pure local computation — no key, no third-party service, instant. Live, nothing stored. 3 endpoints. This is odds conversion; for probability distributions use a probability API.

api.oanor.com/odds-api

Probability distributions and combinatorics as an API. The binomial endpoint gives the probability of exactly k successes in n trials (PMF), the cumulative probability up to k (CDF), and the mean, variance and standard deviation. The poisson endpoint does the same for the Poisson distribution from a rate λ. The normal endpoint computes the z-score, probability density, cumulative probability (CDF) and percentile for a value under a normal distribution with any mean and standard deviation — and runs in reverse, turning a probability into the value (the quantile / inverse CDF) and its z-score. The combinatorics endpoint computes combinations (nCr), permutations (nPr) and factorials with exact big-integer arithmetic. Everything is computed locally and deterministically, so it is instant and private. Ideal for data science and statistics, quality control and A/B-test planning, gaming and gambling odds, risk modelling, and statistics education. Pure local computation — no key, no third-party service, instant. Live, nothing stored. 5 endpoints. This is probability theory; for descriptive statistics on a dataset use a statistics API and for general expression evaluation use a math API.

api.oanor.com/probability-api