# Odds converter (fractional, decimal, American, probability)

> Convert odds between fractional, decimal and American (moneyline) formats and get the implied probability, worked in exact fractions.

Interactive version: https://www.calcopenly.com/statistics/odds-converter
Subject: Statistics and probability calculators

Decimal odds d are the total return per 1 staked, stake included. Fractional odds a/b give the profit per stake, so d = 1 + a/b. American odds are +100(d − 1) when d is 2 or more and −100/(d − 1) when d is below 2, and the implied probability is 1/d. Every conversion runs through exact fractions, so 10/11 stays 10/11 instead of becoming 0.909…

The default, 5/2, is decimal 3.5 and American +250: a 1-unit stake returns 3.5 if it wins, 2.5 of it profit, and the price implies a 28.57% chance. A −110 line, common on US point spreads, implies 52.38%, the win rate needed to break even.

Implied probability describes the price, not the true chance. Bookmakers build a margin into their prices, so the implied probabilities of all outcomes in a market add up to more than 100%.

## Inputs

- **I have** (options: Fractional, Decimal, American, Probability)
- **Fractional odds, first number**: 5/2 means win 5 for every 2 staked.
- **Fractional odds, second number**
- **Decimal odds**: Total return per 1 staked, stake included.
- **American odds**: +250: win 250 on a 100 stake. −150: stake 150 to win 100.
- **Probability**

## Results

- Implied probability — main result
- Decimal odds
- Fractional odds
- American odds
- Profit per 1 staked

## Formula

$$
d = 1 + \frac{a}{b},\qquad \text{American} = \begin{cases} +100(d-1) & d \ge 2 \\ -\frac{100}{d-1} & d < 2 \end{cases},\qquad p = \frac1d
$$

## Worked examples

### 5/2 (defaults)

- I have: Fractional
- Fractional odds, first number: 5
- Fractional odds, second number: 2
- **Decimal odds: 3.5**
- **American odds: +250**
- **Implied probability: 28.57%**
- **Profit per 1 staked: 2.5**
- Checked against: d = 1 + 5/2; p = 2/7 = 28.571%; American +100 × 2.5 (standard conversion tables)

### Decimal 1.5

- I have: Decimal
- Decimal odds: 1.5
- **Fractional odds: 1/2**
- **American odds: −200**
- **Implied probability: 66.67%**
- Checked against: d − 1 = 1/2; American −100/0.5 = −200; p = 1/1.5 = 2/3

### American −110

- I have: American
- American odds: -110
- **Decimal odds: 1.9091**
- **Fractional odds: 10/11**
- **Implied probability: 52.38%**
- Checked against: d = 1 + 100/110 = 21/11; p = 11/21 = 52.381% (the standard −110 break-even rate)

### Probability 25%

- I have: Probability
- Probability: 25%
- **Decimal odds: 4**
- **Fractional odds: 3/1**
- **American odds: +300**
- Checked against: d = 1/0.25 = 4; 3/1; +300

### Edge case: evens

- I have: Decimal
- Decimal odds: 2
- **Fractional odds: 1/1**
- **American odds: +100**
- **Implied probability: 50%**
- Checked against: Even money: decimal 2.0 = 1/1 = +100 = 50%

## Questions

### How do you convert fractional odds to decimal odds?

Divide the first number by the second and add 1: d = a/b + 1. So 5/2 becomes 3.5, 1/2 becomes 1.5 and evens (1/1) becomes 2.0. The 1 is the returned stake: fractional odds quote profit only, while decimal odds quote the total return, so a winning 10-unit bet at 5/2 or 3.5 pays back 35 in both cases.

### How do American (moneyline) odds work?

A plus number is the profit on a 100 stake and a minus number is the stake needed to win 100. At +250, 100 staked wins 250 profit (decimal 3.5); at −150, 150 staked wins 100 (decimal 1.667). +100 and −100 both mean evens, and no American odds exist between −100 and +100.

### How do you convert odds to implied probability?

Divide 1 by the decimal odds: p = 1/d. Decimal 3.5 implies 1/3.5 = 28.57%. For fractional odds a/b the shortcut is b/(a + b), so 5/2 gives 2/7. For American −110 it is 110/(110 + 100) = 52.38%, and for +250 it is 100/(250 + 100) = 28.57%.

### What is the overround or bookmaker's margin?

The overround is how far the implied probabilities of every outcome in a market add up past 100%. With both sides of a bet priced at −110, the total is 52.38% + 52.38% = 104.76%, an overround of 4.76% and a bookmaker margin (vig) of 1 − 1/1.0476 = 4.55% of stakes. Dividing each implied probability by the total, 52.38/104.76, removes the margin and gives 50%.

### What do odds-on and odds against mean?

Odds-on means the implied probability is above 50%: fractional odds below 1/1, decimal odds below 2.0 and negative American odds. At 1/2 odds-on (1.5, or −200) you stake 2 to win 1, an implied 66.67%. Odds against is the reverse: 5/2 (3.5, or +250) pays more than the stake and implies less than 50%.

### How accurate is the odds converter?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 5 worked examples whose answers come from independent sources; for example, “5/2 (defaults)” is checked against d = 1 + 5/2; p = 2/7 = 28.571%; American +100 × 2.5 (standard conversion tables).

### Where does the method come from?

Grinstead & Snell, Introduction to Probability (AMS), §1.2 — odds and probability; UK Gambling Commission — how odds work (fractional and decimal).

## Sources

- [Grinstead & Snell, Introduction to Probability (AMS), §1.2 — odds and probability](https://math.dartmouth.edu/~prob/prob/prob.pdf)
- [UK Gambling Commission — how odds work (fractional and decimal)](https://www.gamblingcommission.gov.uk/public-and-players/guide/page/how-odds-work)

_Note: financial information, not professional advice._