The calculations are done for an event with three outcomes, like a football match ending with either 1 (home win), X (draw), or 2 (visitors win), but can be used quite generally.The only condition is that the outcomes, for which odds are given, are mutually exclusive and jointly cover all possible outcomes of the event.

Step 1 - Calculating the bookmaker's profit indicator

First we calculate what could be termed as the bookmaker's profit indicator. It shows how much the bookmaker receives in stakes whenever he makes a payment of $1. If odds[1] is the (European) odds for a home win, the payout from a $1 bet, then 1/odds[1] is the bookmakers price for a payout of $1 in case of a victory for the home team.The price for paying out $1 irrespective of the outcome of the outcome of the game (where you play all three outcomes) is thus:

In this case the profit indicator shows that the bookmaker recevies a stake of $1.181 for every $1 paid out.

Step 2 - Calculating the bookmaker's payout share

If the bookmaker receives the bets in the right proportions, then the payout share (as part of total stakes for that match) for the customers will be

From the payout share, you'll have the profit share for the bookmaker as 100% - payout share.

Step 3 - Calculating the probability for each outcome

For the final step, multiply the bookmaker's price for each of the three outcomes by the payoutshare. This removes the part of the price, which is the bookmaker's profit, and leaves only the pure probability element of the price.

The calculations above show how to obtain the probabilities intrinsic in the odds of the bookmaker. The bookmaker's probabilities may of course not be correct estimates. When you judge the probabilities to be different, is when you have a good opportunity to make money on a bet.