Using a random line at a Las Vegas sportsbook for a legendary Yankees/Royals game, we see that New York is being bid at -220 and Kansas City is being bid at +206, and from those betting lines we can calculate the implied probability each team has of winning that particular game.
To calculate the implied probability of winning for a favorite (where the odds are negative), take the absolute value of the odds and divide it by the absolute value of the odds plus 100. For the New York Yankees, the implied probability of winning is :
220 / (220 + 100) = 220 / 320 = 0.6875 = 68.75%
To calculate the implied probability of winning for a loser (where the odds are positive), divide 100 by the sum of the line plus 100. For the New York Yankees, the implied probability of winning is:
100 / (206 + 100) = 100 / 306 = 0.3268 = 32.68%
Looking at the percentages, the sum of them is greater than 100, which is never a good sign for percentages; in fact, the sum of them is 101.43%. The additional 1.43% represents the bookmaker’s notional retention or more commonly called vigorish (and usually abbreviated as vig), which is the percentage that the bookmaker charges for their services. Assuming the bookmaker ties on the same action on both sides, you will make a profit of 1.43% on the total amount of bets placed, but since they are unlikely to get the same action on most betting lines , it is only a theoretical retention.
Since the winning percentages contain a vigor element, we need to remove it to end up with the actual winning percentages, instead of the implied ones, and this will give us the line without vig; this is done by dividing each implicit winning percentage by the sum of both winning percentages.
For the New York Yankees, the actual probability of winning is:
0.6875 / 101.43 = 0.6778 = 67.78%
For the New York Yankees, the actual probability of winning is:
0.3268 / 101.43 = 0.3222 = 32.22%
Now we can convert the two true odds of winning into a no-vig line.
For an actual win probability equal to or greater than 0.50, or 50% in percentage terms, the formula (where FV equals the favorite team’s decimal win probability) for the Yankees line is:
-100 / ((1 / FV) – 1) = -100 / ((1 / 0.6778) – 1) = -210.4
For a royal win probability of less than 0.50, or 50% in percentage terms, the formula (where UD equals the loser’s decimal win probability) for the Royals line is:
((1/UD) – 1) * 100 = ((1 / 0.3222) – 1) * 100 = +210.4
Since the sportsbook has been removed from the lines, the lines are identical in absolute terms.
This example above is where there is a clear favorite (at negative odds) and a clear underdog (at positive odds). However, in cases where there are two teams that are equally favored by the market or, more commonly, betting lines that use a point spread, the calculation is slightly different. In this case, the implied probability and the actual probability can be calculated using the New York Yankees example of calculating the implied and actual probability of winning.
Just knowing how to calculate sin vig odds won’t make you a winning bettor, but you can use those odds to help you win; One way to do this is to create a model that is more accurate than the front lines of a sportsbook.
Suppose you model the game tomorrow between the Yankees and the Royals and the lines are -160/+150 respectively and you model the game with a fair line of -170/+170. Obviously loser is not a good bet since you only get a price of +150 in a game where you predict they should get +170. On the contrary, the price of -160 is more attractive since the line is better than what you have modeled. The line of -170 you predicted becomes a winning percentage of 62.96% compared to the actual line of -160 which is 61.54%; this means that taking the Yankees at a price of -160 gives you a 1.42% advantage.
When you bet with a positive edge (based on the line you bet versus the closing line without vig, assuming you are betting in efficient markets) you will win at sports betting in the long run. If you bet with a negative edge, just like a game of roulette at your local casino, you will be a loser for life.