A reference on games of chance
This site explains the subject rather than the industry: how probability and expected value are computed, where a house edge comes from and why it is fixed before anyone plays, how wheels, dice, cards, reel machines, lotteries and betting markets are actually constructed, and how the regulation of all of it developed.
It names no operators, carries no offers, ranks nothing, and has nothing to play. Every number below is worked out in the open from stated assumptions.
Foundations4 entries
- Probability, odds, expectationp = n/N
- House edge and holdedge = -E
- Variance and streakssd = √var
- The long runedge x handle
Wheel and dice games4 entries
- Roulettepockets 37|38
- Dice and crapsoutcomes 36
- Why one edge covers many bets-1/37
- Free odds and zero-edge betsE = 0
Card and machine games4 entries
- Card games and dependencedependent
- Reel machinesRTP != vol
- Composition and memorydeck state
- Virtual reels and weightingstops != symbols
Betting markets4 entries
- Odds notation and overround∑p > 1
- Draw lotteriesC(49,6)
- Balancing a bookliability
- Pari-mutuel poolspool share
Long-run behaviour4 entries
- Law of large numbersn → ∞
- Recycling and handleturnover
- The gambler's fallacyindependent
- Ruin and finite bankrollsabsorbing
History and regulation4 entries
- History of games of chance3000 BCE–
- How regulation developedlicence
- The founding correspondence1654
- What an audit checksRNG + RTP
Reference4 entries
- Glossary of termsA–Z
- Return to playerRTP = 1-edge
- Hold percentagehold
- Independence of trialsi.i.d.
Reading order4 entries
- Start with the arithmeticstep 1
- Then a single gamestep 2
- Then the market sidestep 3
- Then the long runstep 4
Core definitions
probabilityp(A) = favourable / total0 ≤ p ≤ 1
A number between zero and one giving the share of equally likely outcomes in which an event occurs. It is not a prediction about the next trial; it is a statement about the whole set of outcomes the trial is drawn from. Every figure on this site starts here, by counting the outcomes a game actually has.
| field | value |
|---|---|
| outcome set | must be fully enumerable |
| range | 0 to 1 inclusive |
| certainty | p = 1 |
| impossibility | p = 0 |
oddsodds = p : (1 - p)3 forms
A ratio rather than a share. Fractional odds compare the profit to the stake, decimal odds give the total return per unit staked including the stake, and moneyline odds express the same thing relative to a hundred units. All three carry identical information; only the arithmetic of converting them differs.
| field | value |
|---|---|
| fractional | 5/2 |
| decimal | 3.50 |
| moneyline | +250 |
| implied p | 0.2857 |
expected valueE[X] = ∑ xₕ pₕper unit staked
The average result of a wager if it could be repeated indefinitely under the same conditions. It is the single most useful number in this subject, because it does not describe any individual outcome and does not need to: it describes the arithmetic the game is built on.
| field | value |
|---|---|
| fair game | E = 0 |
| commercial game | E < 0 |
| units | per unit staked |
house edgeedge = -E[X] per unit2.70% (single zero)
The negative of expected value, stated as a percentage of the amount staked. It is a property of the payout table and the outcome count, fixed before anyone plays, and it does not vary with how a player feels about a run of results.
| field | value |
|---|---|
| source | payout table vs outcome count |
| single-zero wheel | 1/37 = 2.70% |
| double-zero wheel | 2/38 = 5.26% |
return to playerRTP = 1 - edgeexpressed as %
The same quantity as house edge, seen from the other side, and usually the figure quoted for machine games. An RTP of 96% and a house edge of 4% are one number written two ways. It is a long-run average over an enormous number of rounds, not a description of a session.
| field | value |
|---|---|
| relationship | RTP + edge = 100% |
| horizon | millions of rounds |
| session meaning | none in isolation |
variancevar = E[X²] - (E[X])²sd = √var
The spread of results around the expected value. Two games can share an edge and behave nothing alike: one grinds steadily, the other pays rarely and largely. Variance is what makes short runs uninformative, and it is the reason a losing arithmetic can still produce winning evenings.
| field | value |
|---|---|
| even-money wheel bet | var ≈ 0.999 |
| long-shot bet | var large |
| scaling | sd grows as √n |
overround∑ implied p over all outcomes> 1 by construction
In a betting market the implied probabilities of every outcome sum to more than one. The excess is the margin built into the prices. It is the market equivalent of house edge, and it is visible to anyone willing to convert the quoted odds back into probabilities and add them up.
| field | value |
|---|---|
| fair book | ∑p = 1.000 |
| commercial book | ∑p > 1.000 |
| margin | 1 - 1/∑p |
independencep(A and B) = p(A) × p(B)i.i.d.
Two trials are independent when the result of one tells you nothing about the other. A wheel and a pair of dice have no memory; a shoe of cards does, because removing a card changes what remains. Almost every popular misunderstanding of chance is a failure to check which of those two situations applies.
| field | value |
|---|---|
| wheel | independent |
| dice | independent |
| card shoe | dependent |
| draw lottery | independent between draws |
Worked examples
Two calculations that between them account for most of what follows. The first is where a house edge comes from; the second is where a lottery jackpot probability comes from.
outcomes on a single-zero wheel 37 (0 and 1 to 36)
an even-money bet covers 18 pays +1 unit
it does not cover 19 pays -1 unit
E[unit] = ( 18 x (+1) + 19 x (-1) ) / 37
= ( 18 - 19 ) / 37
= -1 / 37
= -0.027027...
house edge = 2.7027% -> RTP = 97.2973%ways to choose 6 numbers from 49, order ignored:
C(49,6) = 49! / (6! x 43!)
= (49 x 48 x 47 x 46 x 45 x 44) / (6 x 5 x 4 x 3 x 2 x 1)
= 10068347520 / 720
= 13983816
p(one line matches all six) = 1 / 13983816
= 0.0000000715...Scope of this edition
Thirteen entries are published. Ten carry worked arithmetic — foundations, house edge, roulette, dice, card games, reel machines, lotteries, betting markets, variance and the long run. Two are narrative: the history of games of chance and the development of regulation. The glossary indexes the terms the entries use.
Deliberately out of scope, and not planned: operator names, prices, promotions, ranked comparisons, strategy claimed to beat a negative expectation, and any figure that cannot be derived on the page from a stated rule set.
Last modified 17 August 2026.