Betting – Probability and Odds Computing

We start with easy example how bookmakers compute odds. Let us imagine that we are a bookmaker and we want to compute odds for the game where gamblers can bet on the result of a dice throw. In this case, we do not have any problem to compute probabilities of each result. For all results, i.e., 1, 2, 3, 4, 5 and 6 the probability is the same and it is equal to 1/6.

Fair Odds

Fair odds means that a bookmaker does not have any profit margin. In this case it is easy to compute odds while it is reciprocal value of the probabilities. Therefore, for all possible results we give to gamblers “fair odds” equal to 6. Let us assume that our gambler bets 1 USD to the number 2.

• If the result of a dice throw is not number 2, he loses his stake.
• If the result of a dice throw is number 2 we have to pay gambler 6 USD for the winning stake (this means he has 5 USD, i.e., -1 + 6).

This odds are not common in the reality. More common are sub-fair odds.

Sub-Fair Odds

Sub-fair odds are common in reality. This time, bookmaker decides to have a positive profit margin, usually between 5% and 15%. We can resume in the example with dice throw. Now, we assume that a bookmaker has decided to work with 10% profit margin. To compute odds we divide (1-margin), e.g. 0.9 for 10% margin, by the probability of each variant. In our case, we get (1-0.1) / (1/6). The result is 5.4, i.e., for each dollar placed on the winning number we pay back 5.4 USD.

For a bookmaker it means that in 5/6 cases he wins a dollar and in 1/6 cases he losses 4.4 USD (net loss, i.e., 5.4 – 1). If we count mean value out of this we get (+1) * (5/6) + (-4.4) * (1/6) = 0.1.

The previous result means that we can be lucky sometimes but in the long term period we have to loose.

• In 5/6 cases we loose 1 USD and
• In 1/6 cases we win 4.4 USD (-1 USD for the bet and +5.4 from the bookmaker for the winning stake).

Probability Computing

Using approach described above we can compute probabilities using odds from a bookmaker. The first step is to compute profit margin of the bookmaker. To demonstrate the procedure we use real odds from Swedish Elitserien. On December 20, 2012 we recorded odds given by Bwin.com for a match between Färjestads BK and Linköpings HC (played in Färjestads).

• Odd for Färjestads BK is 1.95, i.e., for a 1 USD stake placed on Färjestads BK we get back 1.95 (net gain 0.95) if the Färjestads BK wins.
• Odd for Linköpings HC is 3.05.
• Odd for tie game is 3.85.

Profit Margin Computing and Probabilities Computing

Given odds above we compute reciprocal value of each odd and sum over them.

• 1/1.95 + 1/3.05 + 1/3.85 = 1.10043

To compute profit margin we subtract reciprocal value of the result above from 1, i.e.,

•  1 – 1/1.10043 = 0.09126.

This means that for this match Bwin.com has 9.126% profit margin.

Again, to compute probabilities, we need to use result above, i.e., 1.10043. To compute probabilities (not the real probabilities, but the probabilities assumed by the bookmaker) we easily divide reciprocal value of each odd by 1.10043.

• Probability for win of Färjestads BK is (1/1.95) / 1.0043 = 46.60%.
• Probability for win of Linköpings HC is (1/3.05) / 1.0043 = 29.80%.
• Probability for tie game is (1/3.85) / 1.0043 = 23.60%.

If these probabilities are correct then bookmaker cannot lose in long-term period while mean value of payment for each variant is 0.90874, i.e., 1 – (profit margin).

• For Färjestads BK we get 0.4660 * 1.95 = 0.90874.
• For Linköpings HC we get 0.2980 * 3.05 = 0.90874.
• For the tie game we get 0.2360 * 3.85 =  0.90874.

The problem for the bookmaker arises when he uses bad probabilities. We continue in the previous example but we assume that real probabilities are

• For Färjestads BK we assume 60%.
• For Linköpings HC we assume 20%.
• For the tie game we assume 20%.

As we can see, in our example we have higher probability for the win of Färjestads BK. Other two variants has lower probabilities. This time, the mean value of payment for variants differ themselves and they may not be less than 1.

• For Färjestads BK we get 0.60 * 1.95 = 1.17.
• For Linköpings HC we get 0.20 * 3.05 = 0.61.
• For the tie game we get 0.20 * 3.85 =  0.77.

As we can see, we obtained two values that are better for the bookmaker and one value (1.17) that will be problematic for a bookmaker. The good way how to gain some money in this case is to bet on Färjestads BK, while the mean value is over 1. In this case, the bookmaker can be beaten.

It seams easy, but to compute probabilities is very hard job. The basic information can be found in following academic papers:

• Dixon, M. J., & Coles, S. G. (1997). Modelling Association Footbal Scores and Inefficiencies in the Football Betting Market. Journal of the Royal Statistical Society. Series C (Applied Statistics), 46 (2), 265-280.
• Dobson, S., & Goddard, J. (2011). The Economics of Football. Cambridge University Press, New York.
• Famoye, F. (2010). A new bivariate generalized Poisson distribution. Statistica Neerlandica, 64 (1), 112-124.
• Genest, C., & Neslehova, J. (1997). A Primer for Copulas for Count Data. Astin Bulletin, 37 (2), 475-515.
• Karlis, D., & Ntzoufras, I. (2003). Analysis of sports data by using bivariate Poisson models. The Statistician, 52 (3), 381-393.
• Maher, M. J. (1982). Modelling association football scores. Statistica Neerlandica, 36 (3), 109-118.
• Marek, P., Šedivá, B., Ťoupal, T. (2014) Modeling and prediction of ice hockey match results. Journal of Quantitative Analysis in Sports, 10, (3), 357-365. ISSN: 1559-0410

Reading of these paper requires knowledge of probability, statistics and mathematics at the university level.