---
title: "The Mathematics of Odds"
url: https://betongood.foundation/odds/
description: "Sample spaces, odds notation, expected value and why the arithmetic settles the question."
---

# The Mathematics of Odds

[Bet On Good](https://betongood.foundation/)

Notes on chance, odds and the design of games

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The Mathematics of Odds

The Mathematics of Odds

Everything in this category is arithmetic. A game of chance has a finite list of outcomes and a rule that says what each one pays, and once both are written down there is nothing left to argue about. These entries set out the counting, the notation used to express it, and the single calculation that decides whether a wager is worth more or less than the stake placed on it.

Entries in this category

*Entries filed under The Mathematics of Odds.*

| Entry type | Entry | Last updated | Sections | Figures |
|---|---|---|---|---|
| worked arithmetic | [Probability, odds and the gap between them](https://betongood.foundation/odds/probability-and-odds/)Probability counts outcomes; odds compare them. The two describe the same situation and convert into each other exactly, so any disagreement between a stated probability and a stated price is a fact about the price, not about chance. | 17 August 2026 | 4 | 1 |
| worked arithmetic | [House edge and expected value](https://betongood.foundation/odds/house-edge-and-expected-value/)Expected value multiplies each outcome by its probability and adds the results. For a commercial game the sum is negative by construction, and the size of that negative number is the house edge. | 17 August 2026 | 4 | 1 |
| worked arithmetic | [Variance, the long run and the law of large numbers](https://betongood.foundation/odds/variance-and-the-long-run/)Expectation says where the average goes. Variance says how far individual results scatter around it, and how quickly that scatter stops mattering as the number of wagers grows. | 17 August 2026 | 4 | 1 |

Sections within these entries

[Probability, odds and the gap between them](https://betongood.foundation/odds/probability-and-odds/)

arithmetic

- [Counting the outcomes](https://betongood.foundation/odds/probability-and-odds/#s1)
- [Odds are a ratio, not a fraction](https://betongood.foundation/odds/probability-and-odds/#s2)
- [Where the gap opens](https://betongood.foundation/odds/probability-and-odds/#s3)
- [Independence and the gambler's fallacy](https://betongood.foundation/odds/probability-and-odds/#s4)

[House edge and expected value](https://betongood.foundation/odds/house-edge-and-expected-value/)

arithmetic

- [The calculation](https://betongood.foundation/odds/house-edge-and-expected-value/#s1)
- [Why the number does not change with the bet chosen](https://betongood.foundation/odds/house-edge-and-expected-value/#s2)
- [Edge, hold and the volume of play](https://betongood.foundation/odds/house-edge-and-expected-value/#s3)
- [The one result that follows](https://betongood.foundation/odds/house-edge-and-expected-value/#s4)

[Variance, the long run and the law of large numbers](https://betongood.foundation/odds/variance-and-the-long-run/)

arithmetic

- [Two numbers, not one](https://betongood.foundation/odds/variance-and-the-long-run/#s1)
- [What the law of large numbers actually says](https://betongood.foundation/odds/variance-and-the-long-run/#s2)
- [Why a win proves nothing](https://betongood.foundation/odds/variance-and-the-long-run/#s3)
- [Risk of ruin](https://betongood.foundation/odds/variance-and-the-long-run/#s4)

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[The Mathematics of Odds](https://betongood.foundation/odds/)

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[Rules and Regulation](https://betongood.foundation/regulation/)

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Entry carries worked arithmetic with the steps shown.

Entry is narrative: history, development or design reasoning.

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