# What are the no-vig odds?

Removes the bookmaker’s margin (vig) from the odds of every outcome of one event, and gives each outcome’s fair probability and fair odds, with the size of the margin.

- Page: https://www.acalculator.org/game/no-vig-calculator
- JSON spec: https://www.acalculator.org/game/no-vig-calculator.json
- Version: ce07c6fe5ff9

## Default answer

Example with the default inputs (Odds format American, Outcomes [Odds -150; Odds +130]): With the bookmaker’s margin removed, outcome 1 has a fair probability of 57.98% (-138).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| format | Odds format | The format of every outcome’s odds: American, decimal, or fractional. |
| outcomes | Outcomes | The odds of each outcome of one event from one bookmaker, one row per outcome. List every outcome. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| fair1 | Fair probability, outcome 1 | Outcome 1’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds1 | Fair American odds, outcome 1 | Outcome 1’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal1 | Fair decimal odds, outcome 1 | Outcome 1’s odds with the margin removed: 1 ÷ fair probability. |
| fair2 | Fair probability, outcome 2 | Outcome 2’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds2 | Fair American odds, outcome 2 | Outcome 2’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal2 | Fair decimal odds, outcome 2 | Outcome 2’s odds with the margin removed: 1 ÷ fair probability. |
| fair3 | Fair probability, outcome 3 | Outcome 3’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds3 | Fair American odds, outcome 3 | Outcome 3’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal3 | Fair decimal odds, outcome 3 | Outcome 3’s odds with the margin removed: 1 ÷ fair probability. |
| fair4 | Fair probability, outcome 4 | Outcome 4’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds4 | Fair American odds, outcome 4 | Outcome 4’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal4 | Fair decimal odds, outcome 4 | Outcome 4’s odds with the margin removed: 1 ÷ fair probability. |
| fair5 | Fair probability, outcome 5 | Outcome 5’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds5 | Fair American odds, outcome 5 | Outcome 5’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal5 | Fair decimal odds, outcome 5 | Outcome 5’s odds with the margin removed: 1 ÷ fair probability. |
| fair6 | Fair probability, outcome 6 | Outcome 6’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds6 | Fair American odds, outcome 6 | Outcome 6’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal6 | Fair decimal odds, outcome 6 | Outcome 6’s odds with the margin removed: 1 ÷ fair probability. |
| fair7 | Fair probability, outcome 7 | Outcome 7’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds7 | Fair American odds, outcome 7 | Outcome 7’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal7 | Fair decimal odds, outcome 7 | Outcome 7’s odds with the margin removed: 1 ÷ fair probability. |
| fair8 | Fair probability, outcome 8 | Outcome 8’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds8 | Fair American odds, outcome 8 | Outcome 8’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal8 | Fair decimal odds, outcome 8 | Outcome 8’s odds with the margin removed: 1 ÷ fair probability. |
| fair9 | Fair probability, outcome 9 | Outcome 9’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds9 | Fair American odds, outcome 9 | Outcome 9’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal9 | Fair decimal odds, outcome 9 | Outcome 9’s odds with the margin removed: 1 ÷ fair probability. |
| fair10 | Fair probability, outcome 10 | Outcome 10’s implied probability ÷ the sum of all implied probabilities. |
| fairOdds10 | Fair American odds, outcome 10 | Outcome 10’s odds with the margin removed, in American format, rounded to 2 decimals. |
| fairDecimal10 | Fair decimal odds, outcome 10 | Outcome 10’s odds with the margin removed: 1 ÷ fair probability. |
| overround | Overround | How far the implied probabilities of all outcomes add up past 100%. |
| note | Note | Shown when the implied probabilities add up to 100% or less, which a single bookmaker’s full market never does. |
| vig | Vig (bookmaker’s margin) | The share of all money staked the bookmaker keeps when every outcome pays the same: 1 − 1 ÷ (sum of implied probabilities). |

## Method

With decimal odds dᵢ: implied probability pᵢ = 1/dᵢ; book B = Σ pᵢ; fair probability qᵢ = pᵢ ÷ B; fair decimal odds = 1 ÷ qᵢ = dᵢ × B; overround = B − 1; vig = 1 − 1 ÷ B.

## Assumptions

- The margin is removed in proportion to each outcome’s implied probability (the multiplicative or normalization method). Other methods (additive, power, Shin) give slightly different fair odds, most of all for long shots.
- The odds come from one bookmaker and cover every outcome of the event.
- Fair odds estimate what the market expects; they are not the true chances.
- Results are mathematics, not betting advice.

## Worked examples

1. format = american, outcomes = {"odds":"-110"} or {"odds":"-110"} gives fair1 = 50%, fair2 = 50%, fairOdds1 = +100, fairDecimal2 = 2, overround = 4.761905%, vig = 4.545455%. Source: hand calculation in content.mdx: 11/21 each, sum 22/21; fair 1/2 each.
2. format = american, outcomes = {"odds":"-150"} or {"odds":"+130"} gives fair1 = 57.983193%, fair2 = 42.016807%, fairOdds1 = -138, fairOdds2 = +138, fairDecimal1 = 1.724638. Source: hand calculation in content.mdx: 3/5 ÷ 119/115 = 69/119; 10/23 ÷ 119/115 = 50/119.
3. format = decimal, outcomes = {"odds":"2.2"} or {"odds":"3.4"} gives fair1 = 44.283647%, fairDecimal3 = 3.695187. Source: hand calculation in content.mdx: 5/11 + 5/17 + 5/18 = 3455/3366; fair 1 = 5/11 ÷ 3455/3366 = 306/691.

## FAQ

### What is the vig?

The vig (vigorish, juice or margin) is the bookmaker’s built-in edge. It makes the implied probabilities of all outcomes add up to more than 100%. At −110 on both sides they add up to 104.76%.

### How do I remove the vig from odds?

Turn each outcome’s odds into an implied probability (1 ÷ decimal odds), add them up, and divide each one by the sum. For −150 / +130, the implied probabilities 60% and 43.48% add up to 103.48%, so the fair probabilities are 57.98% and 42.02%.

### What are the fair odds of −110 / −110?

Each side implies 52.38%. Divided by the 104.76% sum, each is 50%, which is +100 (decimal 2.0) in fair odds.

### What is the difference between the overround and the vig?

Both measure the same margin. The overround is how far the implied probabilities add up past 100% (4.76% at −110 / −110). The vig here is the share of the money staked that the bookmaker keeps when every outcome pays the same (1/22 = 4.55%).

### Which method does this calculator use?

The proportional (multiplicative) method: every implied probability is scaled by the same factor. It is the most common method. Others (additive, power, Shin) remove more margin from long shots, so their fair odds for long shots differ.

## Sources

- Wikipedia, Fixed-odds betting (overround): https://en.wikipedia.org/wiki/Fixed-odds_betting
- Grinstead and Snell, Introduction to Probability, 2nd ed., American Mathematical Society, section 1.2, odds and probabilities (page 28): https://math.dartmouth.edu/~prob/prob/prob.pdf
- National Council on Problem Gambling, help and treatment (1-800-MY-RESET): https://www.ncpgambling.org/help-treatment/