In order to profit long term from gambling, it is essential to understand the underlying mathematical concept behind the odds offered by sportsbooks.

Using a random line at Pinnacle Sports for a mythical Yankees/Royals game we see New York offered at -220 and Kansas City 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 that 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 an underdog (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 over 100 which is never a good sign for percentages; in fact, the sum of them is 101.43%. The additional 1.43% represents the theoretical hold for the sportsbook or more commonly called the vigorish (and generally shortened to vig) which is the % amount charged by the sportsbook for its services. Assuming that the sportsbook draws in equal action on both sides it will then make 1.43% profit on the total amount of bets placed but since they are unlikely to attain equal action in most betting lines, it is only a theoretical hold.

Since the winning percentages contain an element of vigorish, we need to remove that in order to end up with the actual, rather than the implied, winning percentages and this will give us the no vig line; this is done by dividing each implied 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 actual win probabilities into a no-vig line.

For an actual win probability equal or greater than 0.50 – or 50% in percentage terms – the formula (where FV is equal to the decimal win probability of the favored team) for the Yankees line is:

-100 / ((1 / FV) – 1) = -100 / ((1 / 0.6778) – 1) = -210.4

For an actual win probability less than 0.50 – or 50% in percentage terms – the formula (where UD is equal to the decimal win probability of the underdog) for the Royals line is:

((1 / UD) – 1) * 100 = ((1 / 0.3222) – 1) * 100 = +210.4

Since the sportsbook vig has been removed from the lines, the lines are identical in absolute terms.

This above example is where there is a clear favorite (with negative odds) and a clear underdog (with positive odds). However in the cases where there are two teams which are similarly favored by the market or, more commonly, the betting lines which use a point spread the calculation is slightly different. In this case the implied probability and actual probability can be calculated by using the New York Yankees example of calculating the implied and actual probability of winning.