Gambling Mathematics
Every gambling game comes down to a few numbers: expected value, house edge, variance. We explain them in plain words, we compute them on real examples — crash games, roulette, slot machines — and we give you the calculator to redo the maths with your own stakes.
Key takeaways
- This page explains gambling mathematics — expected value, house edge, the law of large numbers, variance — so you know the average price of every decision before you play, instead of learning it by losing.
- Expected value is the average outcome of a repeated decision: on casino games it is always negative — it is the average price of the game, computable before the first stake.
- The house edge is the gap between 100% and the RTP: 2% on Chicken Road, 4.5% on Chicken Road 2, 2.7% at European roulette. It applies to the total amount wagered, not to the deposit.
- The law of large numbers says frequencies converge over millions of rounds — and that a personal session can neither confirm nor contradict those percentages.
- Variance decides what you feel: at equal expected value, targeting 10x instead of 1.50x multiplies the standard deviation of your results roughly fourfold.
- 18+ — gambling carries risks of financial loss and addiction: set your limits before you play.
Expected value: the number behind every game
Expected value answers a simple question: if I repeat the same decision a very large number of times, what average outcome do I get per round? You compute it by multiplying each possible outcome by its probability, then adding everything up. No heavy algebra needed: a crash game has only two outcomes — reach your target, or watch the round stop before it.
A complete example, with figures we measured on crash games. You stake 10 MAD and cash out at 2.00x. Over one million measured rounds, 48.8% reach 2.00x (you win 10 MAD net) and 51.2% stop before it (you lose your 10 MAD). Expected value = 0.488 × (+10) + 0.512 × (−10) = −0.24 MAD per round. Repeated, that decision costs on average 24 cents per round — before strategy even enters the picture.
That negative number is not a flaw in the game: it is its business model, and it is the same for every player. What a player chooses — target, stake — changes the distribution of outcomes, never the sign of the expectation (see variance, below). To apply this calculation to the full mechanics of a crash game, read our guide to crash game mechanics.
Redo the calculation with your own numbers:
DEMONSTRATOR — this tool illustrates the mathematics of gambling. It does not predict your results.
The target changes the probability, never the expectation: at the same house edge, cashing out at 1.50x or at 10x costs the same percentage of the stake.
Educational tool — it illustrates the mathematics of the game; it does not predict your results. 18+.
House edge and return to player
Return to player — RTP — is the share of stakes a game pays back to players on average. The house edge is the same gap seen from the other side: 100% minus the RTP. An RTP of 95.5% means a house edge of 4.5%: for every 100 MAD wagered in total, the game keeps 4.50 MAD on average.
| Game | Published RTP | House edge | |
|---|---|---|---|
| Chicken Road (InOut Games) | 98% | 2% | |
| European roulette | 97.3% | 2.7% (a single zero on 37 pockets) | 27 MAD |
| Chicken Road 2 (InOut Games) | 95.5% | 4.5% | |
| Online slot machines | 92 to 97% depending on the title | 3 to 8% | 30 to 80 MAD |
The edge applies to the total volume wagered, not to the deposit: deposit 200 MAD and re-stake it five times and you have wagered 1,000 MAD — on Chicken Road 2, an expected average loss of 45 MAD. That is why a long session costs more than the deposit amount suggests. At a constant 10 MAD stake, 100 rounds are 1,000 MAD wagered: the expected loss is 20 MAD on Chicken Road, 45 MAD on Chicken Road 2 — on average, with wide deviations around it (see variance, below).
RTP in detail, game by game and source by source: our study Cash or Crash: the RTP of crash games. The estate's RTP simulator is published in French: Simulateur RTP.
The law of large numbers: why results converge
The law of large numbers says this: repeat an independent draw and the observed frequency moves toward the true probability, with the gap narrowing as the number of draws grows. It is what lets casinos run on percentages — and what lets our statistics pages quote frequencies to a tenth of a point.
| Number of rounds | Standard deviation of the observed frequency (rounds below 2.00x, probability 51.2%) |
|---|---|
| 100 | ≈ 5 points |
| 10,000 | ≈ 0.5 point |
| 1,000,000 | ≈ 0.05 point |
In practice: after 100 rounds, seeing 45% or 57% of rounds below 2.00x is perfectly ordinary — the sample is too small to show anything. After one million rounds, the measured frequency — 51.2% — is stable to a tenth of a point: that is the sample behind our crash games statistics 2026.
The direct consequence: your session cannot test the percentages. It is too short to measure anything; it can only cost or pay out, around an average it never reveals.
Variance: what a player feels from one session to the next
Variance measures how widely outcomes spread around the expected value. Its standard deviation tells you how far a typical round lands from the average. Two decisions can carry exactly the same expected value and feel completely different: variance is what makes the difference.
Same game (4.5% house edge), same stake (10 MAD), two targets. Cash out at 1.50x: you win in 63.7% of rounds, and the standard deviation of one round is about 7 MAD. Cash out at 10x: you win in 9.6% of rounds, and the standard deviation rises to about 29 MAD. The expected value is identical — −0.45 MAD per round — but over 100 rounds, the standard deviation of the total result goes from about 72 MAD to about 294 MAD, around the same average loss of 45 MAD.
It is variance, not expected value, that manufactures memorable evenings in both directions: the winning streaks that feel like having "figured out" the game, and the losing streaks that push players to chase. It also explains why player testimonials measure nothing: everyone reports one draw from their own session, not the distribution.
The gambler's fallacy and compound probabilities
The gambler's fallacy is the belief that a past outcome makes the opposite outcome "due": after five losing rounds, the sixth feels promised a win. But every round is an independent draw — the game has neither memory nor debt. The next round keeps exactly the same probability, whatever the previous streak looked like.
Compound probabilities, on the other hand, multiply: chaining three cash-outs at 2.00x has a probability of 0.488³, about 11.6%; chaining six losing rounds, 0.512⁶, about 1.8% per six-round sequence. Over a long session, that "rare" event becomes ordinary — that is the calculation that dismantles the martingale, detailed in our guide to crash game mechanics.
The same error runs in reverse: believing a winning streak will continue (the "hot hand"). Both versions lend a memory to a draw that has none. It is also the engine of crash game predictors: selling as a signal what is only a label pasted over ordinary randomness.
What mathematics cannot predict
No formula on this page predicts the next round. Gambling mathematics describes distributions — what happens on average, over very large numbers — and stays silent on the individual draw: the crash point of the round about to start is already fixed, and no calculation reads it.
No staking system changes the outcome either: doubling after a loss, smoothing stakes or following a "signal" redistributes wins and losses across sessions, without ever lifting the expected value above zero on a game with a margin. What mathematics can do is concrete: price the average cost of a decision, size a bankroll for the variance you accept, and expose a marketing argument dressed up as a method.
Mathematics does not say what will happen; it says what each decision costs on average — and that is already enough to decide.
Responsible play
This content is intended exclusively for adults (18+). Gambling involves real risks: financial loss, addiction, isolation. Set time and budget limits before you play, never play to recover a loss, and seek help if gambling stops being a leisure activity. In France, the Autorité nationale des jeux (ANJ) publishes the list of authorised offers and has illegal sites blocked: for a reader in Morocco, it is a concrete reference point for what a regulated offer looks like, even though the framework that applies to them remains the Moroccan one. Check the Moroccan regulations that apply before staking anything online.
To go further
The complete mechanics of a crash game, round by round: Crash game mechanics. The RTP of crash games, measured and sourced: Cash or Crash: the RTP of crash games. The frequencies behind this page: Crash games statistics 2026. Why no predictor can read a draw: Crash game predictors.
Frequently asked questions
What is the difference between RTP and house edge?
It is the same gap seen from two sides: an RTP of 95.5% means the game pays back 95.5% of stakes on average — and keeps 4.5%, which is the house edge. The two numbers describe the same margin.
Does the expected value change with the target multiplier?
No. The probability of reaching the target falls exactly as the multiplier rises: their product — the average return — stays the same, below 100%. What changes with the target is the variance: how often you win, and how wide the swings are.
How many rounds does it take for the percentages to "show"?
Over 100 rounds, the standard deviation of the observed frequency is about 5 points: the true percentage is unreadable. Around 10,000 rounds it falls to about 0.5 point; our measurements cover one million rounds, where it is about 0.05 point.
Can a martingale make the expected value positive?
No. Doubling stakes after each loss redistributes outcomes — many small winning sessions, a few disastrous ones — without changing the expected value of each round, which is fixed by the house edge. The full calculation is in our crash game mechanics guide.
Can mathematics help you win?
It helps you decide, not predict: knowing the average price of each decision, sizing your bankroll, and spotting impossible promises. No method makes the expected value positive on a game with a margin — anyone claiming otherwise is selling something.
References
- InOut Games — published RTP of the Chicken Road (98%) and Chicken Road 2 (95.5%) games and the studio's provably fair documentation.
- Casino.guru — Chicken Road 2 game page (RTP 95.5%), consulted in September 2026.
- Chicken Road Productions — Crash games statistics 2026: frequencies measured over one million rounds (51.2% of rounds below 2.00x; 48.8% reaching 2.00x).
- Autorité nationale des jeux (ANJ) — list of authorised gambling offers in France and blocking of illegal sites — anj.fr