# Odds Notation

## Table of Contents

Suppose I offer you fair odds on whether a fair six-sided die rolls a six. What are those odds? How would I write them down?

This page isn't about what the odds are (go read about probability for
that). This page is about **notation**: how to write those odds
down. Here are some options:

- 5/1
- 1:5 (Probably? See below.)
- 5:1 (Probably? See below.)
- 6.0
- +500
- -20
- five to one against

I think all of these are **the same odds**, written using different
notations.

Below, I will say why I think this, expand on what I think the relationships between each notation-system are, and I'll link to the articles and guides I read when forming these beliefs.

## Wait, why are we doing this?

Yesterday some friends and I were watching a sports match together. One of them had placed a bet, and said he got odds of "4 to 1 ish". I wanted to know what that meant, so I googled the notation.

I couldn't find any single page that explained the relationships between all the different notations that kept getting thrown around. So of course, I had to write one. I got nerd-sniped.

## Equivalences

Let's re-visit the different notations I wrote above. We have a fair six-sided die. We're going to roll it once, and bet on whether or not it comes up a six. In this simple example, we'll assume there's no bookie taking a cut, and the odds accurately reflect the probabilities we're talking about.

In each of our notations, what odds do we get that the dice comes up 6, with probability 1/6? And what odds do we get of not-6 with probability of 5/6?

Notation | Odds of a 6 | Odds of anything else |
---|---|---|

Decimal | 6.0 | 1.167 |

English | 5/1 | 1/5 |

English(?) | 1:5 | 5:1 |

English(?) | 5:1 | 1:5 |

On/Against | five to one against | five to one on |

American | +500 | +20 |

American | -20 | -500 |

In a moment, we'll look at each system in turn, and say a little about what it means. But first, there's some more basic terminology we have to get straight…

## Winnings vs Payout

If I bet 1 gold nugget that the dice will come up 6, and then the dice comes up 6, here's what I want to happen:

- I get my original gold nugget (my "stake") back
- I get five more gold nuggets
**as well**

I could say that "I won five gold nuggets, and I got my stake back". Or I could say "I got a payout of six gold nuggets, including my stake".

I don't know if this terminology is consistent or universal, but on the sites I've been looking at, it seems to be consistent enough. In this page:

- I'm going to use "won" or "winnings" to mean "excluding my original stake".
- I'm going to use "payout" to mean "including my original stake".

As we shall see, some notations make more sense in terms of "winnings" and others make more sense in terms of "payout".

## Decimal (or European)

According to MyBettingSites, this notation is becoming popular because of exchanges like betfair, and because horse racing venues like Cheltenham think they're easier for new punters to understand.

MyBettingSites say that in this notation:

winnings = (odds * stake) – stake

However, I think it's easier to think about this notation in terms
of **payout**:

- payout (if you win) = odds * stake
- payout (if you lose) = 0

In terms of our dice roll, if I bet 1 nugget on the dice coming up 6:

- I should get odds of 6.0
- If I win, I get 6 nuggets in payout
- These 6 nuggets are comprised of my 1-nugget stake, and 5 nuggets of winnings.

## English (or Fractional)

I've come across two different sub-notations for English odds. One
uses a "`/`

" character, the other a "`:`

". In both cases, I think
it's easier to think in terms of **winnings**.

### With a `/`

Let's start with the "`/`

". Sportsbetting say:

If you placed a bet on 3/1 odds and won, you would win 3 units, plus your initial unit bet for a total payout of 4.

So, I think:

- Odds of
**X/Y**mean if I**bet Y**I stand to**win X**.

In terms of our dice roll, if I bet 1 nugget on the dice coming up 6:

- I should get odds of 5/1
- If I win, I get 5 nuggets in winnings
- …plus my original stake of 1 nugget
- …which comes to 6 nuggets payout

### With a `:`

Confusingly, the "`:`

" notation seems to have the exact same
meaning and "`/`

", with the exact opposite notation. Wikipedia says

wagering 1 at 1:2 pays out 3

So, I think:

- Odds of
**X:Y**mean if I**bet X**I stand to**win Y**.

In terms of our dice roll, if I bet 1 nugget on the dice coming up 6:

- I should get odds of 1:5
- If I win, I get 5 nuggets in winnings
- …plus my original stake of 1 nugget
- …which comes to 6 nuggets payout

In Haskell notation, we might say that `(:) = flip (/)`

**EDIT**: This other Wikipedia page seems to use "`:`

" as exactly
like "`/`

":

Odds of 4:1 ("four-to-one" or less commonly "four-to-one against") would imply that the bettor stands to make a £400 profit on a £100 stake.

Maybe the first one I came across is just mistaken? Or maybe there are multiple ways to use the notation. I don't know.

### On/Against

Obviously this business with "`/`

", and "`:`

" is confusing. To
compound the confusion, I suspect that both "`/`

" and "`:`

" can be
pronounced the same as the English-language word "to". I suspect
that the notion of "on" vs "against" evolved to help people
disambiguate between these two contradictory notations in a spoken
context.

According to 888sport:

“Odds on” is the opposite of “odds against”. It means that the event is more likely to happen than not … When the probability that the event will not happen is greater than the probability that it will, then the odds are “against” that event happening.

I think this is exactly what you need to know in order to
disambiguate between `5/1`

and `5:1`

. In the case of `5/1`

, the
odds are **against** you ("5 to 1 against"). In the case of `5:1`

,
the odds are **for** you ("1 to 5 on").

I note that 888sport have strong opinions about the bigger number
always coming first when the odds are against you, and the smaller
number coming first when they're "on". But then 888sport doesn't
ever use the "`:`

" notation found on wikipedia, so I don't know if
that convention holds universally.

So, I think if someone said either of the following:

- "Five to one against"
- "One to five against"

… I'd think they meant exactly the same as "5/1".

Similarly, I think if someone said either of the following:

- "One to five on"
- "Five to one on"

… I'd think they meant exactly the same as "1/5".

## American (or Moneyline)

This is another one that seems to be easiest to understand in terms
of **winnings**. Sportsbetting say:

A plus sign (+) indicates how much will be won in profit for a 100 unit bet. A minus sign (-) shows how much is needed to bet in order to win 100 units.

So I think:

- Odds of +X mean that if I place a 100 gold-nugget bet, and win, then:
- I will get my original stake of 100 gold-nuggets back,
**and** - I will also get X gold nuggets in winnings

- I will get my original stake of 100 gold-nuggets back,
- Odds of -Y mean that in order to get 100 gold nuggets of winnings, I need to bet Y nuggets in the first place.

In terms of our dice roll:

- If I bet
**100**nuggets on the dice coming up 6… - I should get odds of
**+500** - If I win, I get
**500**nuggets in winnings - …plus my original stake of 100 nugget
- …which comes to 600 nuggets payout

…or equivalently, if we divide all the numbers above by 100:

- If I bet 1 nugget on the dice coming up 6…
- I should get odds of +500
- If I win, I get 5 nuggets in winnings
- …plus my original stake of 1 nugget
- …which comes to 6 nuggets payout

…or equivalently again, if we multiply all **those** numbers by 20:

- If I bet
**20**nuggets on the dice coming up 6… - I should get odds of
**-20** - If I win, I get
**100**nuggets in winnings - …plus my original stake of 20 nugget
- …which comes to 120 nuggets payout

## How'd I do?

I only just learned about all this notation, so I could easily have gotten things wrong! If you know of a notation I've missed out, if I've misread one of my sources, or if you know of contradictory sources, then feel free to tweet at me.