# Implied probability

In sports betting implied probability is the conversion of betting odds into percentages. This calculation takes into account the bookmaker's vigorish, and reveals how often a given bet must win to break even. Wagers that are believed to win more often than the implied probability are said to have positive expected value and are used to generate long term profit.

## Contents

Sportsbook house advantage exists through the implied probabilities of events summing to more than 100%. The greater the sum the larger the bookmaker's edge. Implied probability is calculated by dividing the initial risk by the potential return (which includes both risk and profit).

 Team Spread Total Moneyline Green Bay Packers -3.5 (-110) Over 48.5 (-110) -180 Seattle Seahawks +3.5 (-110) Under 48.5 (-110) +160

Consider the spread market in the example above. Green Bay -3.5 and Seattle +3.5 both have the same odds at -110. An initial risk of \$110 on either side would yield \$100 profit for a total return of \$210. Implied probability is calculated by dividing the initial stake (\$110) by the potential return (\$210):

 Bet Odds Initial risk Total return Implied probability Green Bay -3.5 -110 \$110 \$210 \$100 / \$210 = 0.5238 or 52.38% Seattle +3.5 -110 \$110 \$210 \$100 / \$210 = 0.5238 or 52.38%

Both sides of the market have an implied probability of 52.38% which sum to 104.76%. Given that a fair market would sum to 100% this reveals a house advantage of 4.76%. To generate a profit over a large sample size bettors need to win more than 52.38% of their wagers at -110 odds.

## Converting odds formats

Implied probability can always be calculated by dividing the initial risk by the total return. Betting odds directly or indirectly indicate potential payout relative to stake and can therefore be used to derive formulas for calculating implied probability. Below are formulas used to convert popular odds formats into percentages.

### American

American odds represent betting prices through positive and negative numbers relative to a 100 unit base figure. Negative American odds require a risk of that amount in order to profit \$100. Positive American odds yield a profit of that amount for an initial stake of \$100. Converting positive and negative American odds to implied probability require separate formulas.

 Team Spread Total Moneyline New England Patriots -3.5 (-110) Over 44.5 (-110) -180 Baltimore Ravens +3.5 (-110) Under 44.5 (-110) +160

In the NFL match above the New England Patriots are favored with -180 odds to win outright. Implied probability is calculated by dividing initial risk by total return. Since a bet on a market with -180 odds requires a risk of \$180 in order to profit \$100, it is converted into a percentage by diving the odds (180) by the odds augmented by 100 (180 + 100 = 280). The Baltimore Ravens are the underdog with +160 odds to win outright. An initial stake of \$100 would yield \$160 profit and a return of \$260. Converting positive American odds into implied probability is done by dividing the odds (160) by the odds augmented by 100 (160 + 100 = 260).

 Bet Odds Calculation Implied probability Patriots ML -180 180 / (180 + 100) = 0.6429 64.29% Ravens ML +160 100 / (160 + 100) = 0.3846 38.46%

### Fractional

Fractional odds represent profit relative to initial risk. Odds are represented by a fraction. Betting the amount in the fraction's denominator will profit the amount in the numerator. Since implied probability is calculated by dividing initial risk by total return fractional odds can be converted into percentages by dividing the denominator by the sum of the numerator and denominator:

 Team Spread Total Moneyline Los Angeles Lakers -11.5 (10/11) Over 217.5 (10/11) 1/9 Memphis Grizzlies +11.5 (10/11) Under 217.5 (10/11) 6/1

Consider the moneyline market in the NBA game above. The Los Angeles Lakers are favored and have 1/9 fractional odds of winning the match outright. The Memphis Grizzlies are the underdog and are priced at 6/1. Implied probability is calculated by dividing the denominator by the sum of the numerator and denominator.

 Bet Odds Calculation Implied probability Lakers ML 1/9 9 / (1 + 9) = 9/10 = 0.90 90.00% Grizzlies ML 6/1 1 / (6 + 1) = 1/7 = 0.1428 14.28%

### Decimal

Total return relative to initial risk is indicated through decimal odds. Because implied probability is calculated by dividing the stake by the total return this makes it an easy odds format to convert into percentage. Divide 1 by the decimal odds to convert it to implied probability.

 Team Puck line Total Moneyline Montreal Canadiens -1.5 (2.70) Over 6.5 (1.91) 1.66 Toronto Maple Leafs +1.5 (1.50) Under 6.5 (1.91) 2.30

Consider the moneyline market in the example above. Montreal is favored and has 1.66 odds to win the match. As the underdog Toronto has 2.30 odds. Dividing 1 by these decimal odds will convert them to implied probability.

 Bet Odds Calculation Implied probability Montreal ML 1.66 1 / 1.66 = 0.6024 60.24% Toronto ML 2.30 1 / 2.30 = 0.4348 43.48%