Using a random line at a Las Vegas casino bookmaker for a mythical Yankees/Royals game, we see that New York is offered at -220 and Kansas City at +206, and from those betting lines, we can Calculate the implied probability that each team has of winning that special 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 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 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 theoretical retention or more commonly called vigorish (and usually abbreviated as vig), which is the percentage charged by the bookmaker for its services. Assuming the bookmaker draws on the same action on both sides you will make a profit of 1.43% on the total amount of bets placed, but given that they are unlikely to get the same action on most betting lines , it is only a theoretical retention.

Since winning percentages contain an element of vigor, we need to remove it to end up with the actual winning percentages, rather than the implied ones, and this will give us the line without vig; this is done by dividing each implied winning percentage by the sum of both winning percentages.

For the New York Yankees, the true probability of winning is:

0.6875 / 101.43 = 0.6778 = 67.78%

For the New York Yankees, the true probability of winning is:

0.3268 / 101.43 = 0.3222 = 32.22%

Now we can convert the two actual odds of winning into one line without vig.

For a true probability of winning equal to or greater than 0.50 – or 50% in percentage terms – the formula (where FV equals the favored team’s decimal probability of winning) for the Yankees line is:

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

For an actual 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 was removed from the lines, the lines are identical in absolute terms.

This example above is where there is a clear favorite (with negative odds) and a clear underdog (with 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 odds without vig 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 first 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 underdog is not a good bet, as you only get priced at +150 in a game where you predict they should get +170. On the contrary, the -160 price is more attractive since the line is better than what you have modeled. The -170 line you predicted converts to a winning percentage of 62.96% compared to the actual -160 line 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 with no vig, assuming you are betting on efficient markets) you will win in 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.