---
title: "A reference on games of chance"
url: https://sacstat.org/
description: "A reference on games of chance: probability and expected value, house edge, how wheels, dice, cards, machines and lotteries are constructed, and how betting came to be."
---

# 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, expectation](https://sacstat.org/foundations/)p = n/N
- [House edge and hold](https://sacstat.org/house-edge/)edge = -E
- [Variance and streaks](https://sacstat.org/variance/)sd = √var
- [The long run](https://sacstat.org/long-run/)edge x handle

## Field notes

[01**Responsible Gambling Tools: Mistakes and Better Approaches**Sep 14, 2026](https://sacstat.org/what-responsible-gambling-tools-do-common-mistakes-and-better-approaches/)

[02**SweepstakesTable Review (2026): Features, Pricing & Verdict**Aug 20, 2026](https://sacstat.org/sweepstakestable-review-2026-features-pricing-verdict/)

[03**Why Is SweepstakesTable the Best for No Deposit Sweeps Coins?**Aug 19, 2026](https://sacstat.org/why-is-sweepstakestable-the-best-for-no-deposit-sweeps-coins/)

[04**Why Is SweepsPulse the Best for New Players?**Aug 17, 2026](https://sacstat.org/why-is-sweepspulse-the-best-for-new-players/)

[05**Why Is SweepsPulse the Best for Mobile Apps?**Aug 15, 2026](https://sacstat.org/why-is-sweepspulse-the-best-for-mobile-apps/)

[06**5 WhatsApp Bulk Message Sender Tools for Scheduled Campaigns**Aug 12, 2026](https://sacstat.org/5-whatsapp-bulk-message-sender-tools-for-scheduled-campaigns/)

[07**Top 5 Sweepstakes Casinos for Mobile Apps**Aug 11, 2026](https://sacstat.org/top-5-sweepstakes-casinos-for-mobile-apps/)

[08**Best Sweepstakes Casinos for No Deposit Bonus (2026)**Aug 08, 2026](https://sacstat.org/best-sweepstakes-casinos-for-no-deposit-bonus-2026/)

More

[09**Who's the Best for Highly-Scalable Platform: SOFTSWISS vs Kanggiten**Aug 07, 2026](https://sacstat.org/who-s-the-best-for-highly-scalable-platform-softswiss-vs-kanggiten/)

[10**How to Choose the Best Sweepstakes Casinos for Live Dealer Games**Jul 30, 2026](https://sacstat.org/how-to-choose-the-best-sweepstakes-casinos-for-live-dealer-games/)

[11**Top 5 Sweepstakes Casinos for Real Prizes**Jul 30, 2026](https://sacstat.org/top-5-sweepstakes-casinos-for-real-prizes/)

[12**Top 7 Alternatives of FSB Technology**Jul 24, 2026](https://sacstat.org/top-7-alternatives-of-fsb-technology/)

[13**Who's the Best for Sports Betting Solutions: EveryMatrix vs Kanggiten**Jul 22, 2026](https://sacstat.org/who-s-the-best-for-sports-betting-solutions-everymatrix-vs-kanggiten/)

[14**EveryMatrix vs Kanggiten for Modular API-Driven Solutions**Jul 21, 2026](https://sacstat.org/everymatrix-vs-kanggiten-for-modular-api-driven-solutions/)

[15**Who's the Best for Modular API-Driven Solutions: NuxGame vs Kanggiten**Jul 21, 2026](https://sacstat.org/who-s-the-best-for-modular-api-driven-solutions-nuxgame-vs-kanggiten/)

[16**5 WhatsApp Business API Platforms Worth Shortlisting for Bulk Messaging**Jul 16, 2026](https://sacstat.org/5-whatsapp-business-api-platforms-worth-shortlisting-for-bulk-messaging/)

## Wheel and dice games4 entries

- [Roulette](https://sacstat.org/roulette/)pockets 37|38
- [Dice and craps](https://sacstat.org/dice/)outcomes 36
- [Why one edge covers many bets](https://sacstat.org/roulette/#uniform)-1/37
- [Free odds and zero-edge bets](https://sacstat.org/dice/#odds)E = 0

## Card and machine games4 entries

- [Card games and dependence](https://sacstat.org/card-games/)dependent
- [Reel machines](https://sacstat.org/slot-machines/)RTP != vol
- [Composition and memory](https://sacstat.org/card-games/#memory)deck state
- [Virtual reels and weighting](https://sacstat.org/slot-machines/#virtual)stops != symbols

## Betting markets4 entries

- [Odds notation and overround](https://sacstat.org/betting-markets/)∑p > 1
- [Draw lotteries](https://sacstat.org/lotteries/)C(49,6)
- [Balancing a book](https://sacstat.org/betting-markets/#balance)liability
- [Pari-mutuel pools](https://sacstat.org/lotteries/#parimutuel)pool share

## Long-run behaviour4 entries

- [Law of large numbers](https://sacstat.org/long-run/#lln)n → ∞
- [Recycling and handle](https://sacstat.org/long-run/#handle)turnover
- [The gambler's fallacy](https://sacstat.org/variance/#fallacy)independent
- [Ruin and finite bankrolls](https://sacstat.org/long-run/#ruin)absorbing

## History and regulation4 entries

- [History of games of chance](https://sacstat.org/history/)3000 BCE–
- [How regulation developed](https://sacstat.org/regulation/)licence
- [The founding correspondence](https://sacstat.org/history/#pascal)1654
- [What an audit checks](https://sacstat.org/regulation/#audit)RNG + RTP

## Reference4 entries

- [Glossary of terms](https://sacstat.org/glossary/)A–Z
- [Return to player](https://sacstat.org/house-edge/#rtp)RTP = 1-edge
- [Hold percentage](https://sacstat.org/house-edge/#hold)hold
- [Independence of trials](https://sacstat.org/variance/#fallacy)i.i.d.

## Reading order4 entries

- [Start with the arithmetic](https://sacstat.org/foundations/)step 1
- [Then a single game](https://sacstat.org/roulette/)step 2
- [Then the market side](https://sacstat.org/betting-markets/)step 3
- [Then the long run](https://sacstat.org/long-run/)step 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.

*Defining values for probability*

| 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.

*Defining values for odds*

| 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.

*Defining values for expected value*

| 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.

*Defining values for house edge*

| 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.

*Defining values for return to player*

| 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.

*Defining values for variance*

| 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.

*Defining values for overround*

| 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.

*Defining values for independence*

| 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.

Expectation of a one-unit even-money bet on a single-zero wheel

```
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%
```

Counting the lines in a 6-from-49 draw

```
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.