# How To Calculate Probabilities For Football Betting - Part 2

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How do we calculate probabilities for betting on football games? Today on the blog Cassini delivers part 2 of his series on the application of Poisson distribution.

## Zeroing In

The intricacies of the Poisson distribution need not be fully understood for you to make use of it, because Microsoft’s Excel has a built-in Poisson function. Before we look at Poisson in action, it is important to know that several studies have found that the probability of draws is underestimated by Poisson.

The reason for this is that the probability of zero, and to a lesser extent 1, is under-estimated by Poisson, which is why an adjustment needs to be made to the output. A search of the Internet for ‘zero-inflation’ – a fancy term for increasing the probability of zero – will reveal a number of studies, and some ideas on how to apply this.

## Nuts And Bolts

In statistical terms, the formula for Poisson in Excel is POISSON(x, mu, cumulative) where mu is the mean – the estimated probability of that team scoring a goal – i.e. our modified result. The third parameter ‘cumulative’ is set to FALSE, which results in POISSON returning the probability that a random variable, in this case goals, takes on a value exactly equal to x.

In other words, if you want the probability that a team will score 2 goals, and your calculated goal expectancy (modified score) is 2.127, then your formula is POISSON(2,2.127,FALSE). The output from this is .2696 – i.e. there is a 27% probability of this team scoring exactly 2 goals.

You need to know the probability for all goals though, or at least all likely goals. I look for 0,1, 2, 3, 4 and 5, and then have a 6 or more category which is calculated by subtracting the sum of 0 through 5 from 1. A real life example from the upcoming opening fixture of Arsenal v Sunderland where Arsenal’s goal expectancy’s 2.127 and Sunderland’s is 0.75 gives the following:

These are the raw number, and need to be adjusted for the zero-inflation I mentioned earlier. Once this is done, it is a relatively simple process to calculate the odds for most markets.

I set up a table in my spreadsheet showing the estimated probability of all results:

The home team score is by row, the away team score is by column. Want to know the probability of a 0-0draw? It’s in the (0,0) cell – 0.0755. The area in yellow shows the cells that need to be summed to give the odds on a home win, the cells in blue are for an away win, while the green cells show the draw outcomes.

Not sure what the probabilities translate to in terms of odds? Simply set up a table corresponding with the one above, and display the odds in decimal format. Several scores are rather unlikely, but the coloured area usually covers most of the more probable outcomes.

## Market Making

It’s also a simple process to calculate your odds on all the Under / Over markets, e.g. the Under 1.5outcome is the sum of the probabilities of the 0-0, 1-0 and 0-1 score line. The Under 2.5 is those three plus the 1-1, 2-0 and 0-2, and so on.

To continue with the real life example of Arsenal v Sunderland, the Match Odds (H, A, D) work out to1.49, 8.42, 4.79. The Under 2.5 market comes to 1.94. The Betfair market currently has the Under 2.5 at around 2.12 – the market expects more goals than my spreadsheet. The Match Odds market suggests the market has more confidence in Arsenal than my spreadsheet going 1.4, 10 and 5.2 respectively.

## Caution

The value in this example would appear to be backing the Unders – the 2.5 goal market shows a 9.3% edge,but one limitation of objective pricing models is that they don’t take into account close-season changes. Summer signings and departures will take a few games to be fully absorbed by the spreadsheet, so the numbers your computer spits out should always be validated. If the value looks too good to be true,there’s likely a very good reason for that.

And once again, as with the Elo ratings, many of the decisions that ultimately result in your estimate of the odds are subjective and personal, but as I mentioned in Part One of this Poisson series, by reverse engineering back from the market, you can see where you might need to make adjustments. Once you understand how the prices in all markets are derivatives of the goal expectancy numbers, then the easier it is for you to make adjustments.

Read Part 1 of How to Calculate Probabilities For Football Betting.