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Fisherman’s Friend: Can Poisson Distribution Improve Your Betting?

28th February 2019 / Julian
Betting Advice

Don’t you just wish that there was some nice and simple formula that you could apply to your sports betting strategies? Well, we’re not sure if it’s quite as nice or as simple as you hoped, but Poission distribution is a method that many sports bettors have tinkered with in the hope to improve their gambling fortunes, especially in goal markets. Does it work? How does it work? What’s the catch? What have fish got to do with it?

While punters should always be wary of “formulas” and “equations” when it comes to sports gambling, the Poisson distribution method can be a valuable tool when it comes to scouting out the best value odds on the market. It’s also a great method to use when laying your own odds on a betting exchange.


Essentially, this way of betting isn’t going accurately predict the outcome of football matches or who is going to win the Grand National – no mathematical equation in the natural universe could possibly do that – but it can give bettors a better understanding of how odds are formulated. And that in itself is one of the best ways to improve betting success.

Not as fishy as it sounds

If you can recall any of your GCSE French, then you might remember that “poisson” is the word for “fish”. But this theory has nothing to do with marine life, since its name is actually derived from the founder of the probability system – a nineteenth century French mathematician, Simeon Denis Poisson.

We’re not going to lie, the definition of this theory is a little bit frightening: “a discrete frequency distribution which gives the probability of a number of independent events occurring in a fixed time”. Yeah, we also zoned out about half way through reading that definition.

To simplify, the theory can be used to work out how likely it is for multiple things to happen within a set period of time, as long as the thing that happens can be counted in whole numbers and the occurrence of it happening doesn’t directly affect the chances of it happening again. So, we can use Poisson distribution to guess how likely it is that we receive x amount of letters per day, receive x amount of phone calls in an hour, or see a team score x amount of goals in a 90 minute football match.

We don’t know many online sportsbook sites which take wagers on the postman’s rounds but we do know that most bookies offer an array of goal markets for football matches. This includes correct score, over/under goal markets, Asian handicaps and both teams to score. Using the Poisson method, punters can get a mathematical perspective on how likely it is for a team to score a set number of goals in a match, meaning that punters can view these goal markets in a whole new light. And if you can use the formula to predict the scoreline, then you can work out the most likely match result – in theory.

How does it work?

The Poisson distribution formula looks like this: P(x; μ) = (e-μ) (μx) / x!

Scary, right?

Thoughtful confused handsome man has many questions no answer

Fear not, because will show you a little cheat later on. First of all, we need to determine some inputs to feed into the formula. Since the ultimate aim is to turn mean averages into a figure of probability for different outcomes, we need to work out the average number of goals a team will score in a game. We’ll call this the “average rate of success”.

In order to do this we need to select a league (for this example we’ll stick with the Premier League) and a representative range of data. It is important to ensure that we don’t choose a data range that is too large or too small – the 38 games of a Premier League season is a manageable number which should give a good indication of a team average goal stats.

To begin with, we need to calculate the average “attack strength” and “defence strength” across the league by totting up how many goals in total were scored and conceded in both home and away games. Then we simply need to divide the number of goals by the total number of games played.

Looking at a full Premier league table from 2015/16, we can see the following attack strength statistics:

  • 567 goals scored at home / 380 games = average of 1.492.
  • 459 goals scored on the road / 380 games = average of 1.208.

It’s pretty easy to work out the the defence strength since we simply need to flip the averages above. So, the average number of goals conceded by home teams was 1.208 while the average number of goals conceded by away teams was 1.492.

Now that we know the average attacking and defensive records of the league as a whole, we can apply it to an actual fixture. Let’s imagine Arsenal are playing at home against Liverpool. From the previous season’s statistics we can calculate Arsenal’s attack strength at home:

  • Arsenal scored 31 goals in 19 home games for an average of 1.632.
  • Divide this by the overall league average: 1.632/1.492 = 1.094.
  • Arsenal’s home attack strength is 1.094

We can also calculate Liverpool’s defence strength when playing away:

  • Liverpool conceded 28 goals in 19 away games for an average of 1.477.
  • Divide this by the overall league average: 1.477/1.492 = 0.989.
  • Liverpool’s away defence strength is 0.989.

Now, we can apply a formula simple formula to work out the mean number of goals that Arsenal will score: Number of goals = Arsenal attack strength x Liverpool defence strength x average number of goals.

  • g. 1.094 x 0.989 x 1.492 = 1.614

Apply the same working out to Liverpool’s away attack strength and Arsenal’s home defence strength and we can see that the away team are likely to score 0.752 goals.

Applying a Poisson distribution

Now, we’ve never come across a football match that has ended with a scoreline of 1.614 – 0.752. This is where the Poisson distribution comes into effect, because we can treat these numbers as the “average rate of success” in the equation against a series of variables i.e. the number of goals that a team could possible score in a match.

Remember that ugly equation from earlier? Well, you won’t need to worry too much about that since you can use this Poisson distribution calculator to do the working out for you. Simply enter a whole to indicate the potential number of goals that a team will score into the “Poisson random variable” box and then, enter the average number of goals that the relevant teams scores at home or away into the “Average rate of success” box. The “Poisson probability” indicates the likelihood of the numbers goals being scored.

We entered numbers 0 – 5 into the variable categories for both Arsenal and Liverpool and this is what we got:

Goals 0 1 2 3 4 5
Arsenal 19.91% 32.13% 25.93% 13.95% 5.62% 1.82%
Liverpool 47.14% 35.45% 13.33% 3.34% 0.63% 0.09%

Punters can now use these probabilities to work out how likely various scorelines. If we wanted to know how likely a 1-0 home win was, we would use the following multiplication sum: 0.3213 x 0.4714 = 0.1514. So it is 15.14% likely that this game will end up 1-0 to Arsenal.

Reality bites

When Arsenal played Liverpool in the first game of the 2016/17 season, the away team actually won with a 3-4 scoreline. The likelihood of that happening was 0.89%, which just goes to show how ineffective the Poisson distribution method is when it comes to predicting correct scores and even match outcomes.

The truth is that the mathematical equations just don’t factor in the many variable factors which can effect a football match. Is game being played at the beginning or end of the season? Are there any injured players? Did the kit man give the new star striker the wrong type of studs causing him to slip and skew a shot wide when the goal was open?

Still worthwhile?

Despite is shortcomings, punters can use the Poisson distribution method to good effect when it comes to evaluating the odds on the market. Since the formula can effectively tells us the probability of various outcomes in the goal and match result markets, players can use this to their advantage to sniff out where the best betting value lies.


According to our Poisson distribution findings, the odds for Liverpool to win 3-4 at the Emirates were 0.89%. This converts to decimal odds of 112.36 (100/0.89), meaning that we would have ideally looked for odds higher than this. As it turned out, BetVictor offered odds of 126.00 for this correct score bet.

We’ve used a Premier League football match as an example for ease of reference. However, if players really want to use the Poisson distribution method to exploit favourable odds, then they are likely to be more successful in less popular markets such as lower football leagues, foreign leagues and obscure sports. The simple fact of the matter is that bookmakers tend to get their odds spot on when it comes to the major leagues and the saturation of popular markets in big events offers little room to exploit fluctuations.

The tip is based on the personal opinion of the author. No success is guaranteed. Please gamble responsibly. 18+

* All mentioned odds were valid at the time of writing. Betting odds are subject to fluctuations. Please check the current odds with the respective bookmaker!

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