Binomial Probability (Scotland vs Ireland)
In a previous article I calculated the probabilities of a match based on the Poisson distribution and average goals. The Poisson distribution is a mathematical model to describe the probability of the number of events occurring in a set period of time. It is often applied to things like the number of cars arriving at a red light (in traffic management) or the number of claims per week (in insurance industry). It is surprisingly accurate. We applied it to goals per match.
The Binomial Distribution describes another kind of random variable. It is a little more complicated than the Poisson but can also be applied to goals/football. It describes the number of positive outcomes in n trials given the probability of success p in each individual trial. In our case we will think of a shot as a trial, and the outcome is successful if the shot results in a goal. So to apply this we need to estimate the number of shots for a team per game (this will be n), and we will also need to estimate the probability that a shot results in a goal (this will be p). k is the number of goals.
So I looked back on 4 years data focussing on Scotland home matches and Ireland away matches. I picked 4 years because it is the natural cycle of international matches. Also, if I picked any shorter period then the number of matches would be too few to give a realistic long term average for our estimates of n and p. Poisson was a little easier as all we needed was the goals scored and goals conceded which is freely available on any results service. To estimate shots per game and shot success I needed to dig a little deeper and find total number of shots per game for and against. I had little success here but finally found on www.soccerway.com that they did have the data, but only for a limited number of matches in the last 4 years. I made no distinction between friendly and competitive matches.
So now to estimate our Binomial parameters based only on the above information.
n: This is easy. We will average out the number of shots per match. So Scotland average 10.83 shots per match in my sample. Ireland averaged 14.36 shots against. So it would be reasonable to assume that there will be about 12.59 (the midpoint) home shots in this match. By the same process I estimated that there would be about 9.12 shots for Ireland (away shots).
p: This is a little harder to figure out but still straight forward. Of Scotland last 65 shots that I have recorded off soccerway's data they scored 6. So by division the probability of a shot resulting in a goal is 6/56 = 0.0923. This is not the end of the story because its doesn't take into account Ireland's effectiveness as denying shots. So I calculate this too which is 15 goals conceded on the last 201 shots ie 0.0746 probability. Again average these out and it would be reasonable to expect that Scotland will score 8.3% of their shots against Ireland. Similarly if you apply the same process then Ireland will score 7.6% of their shots. We will take these percentages as the probabilities and use as p in the Binomial formula.
To calculate the Probability that Scotland score 0 goals we just fill in the parameters above (n=12.59, p=8.3%) for k=0. We do the same for k=1, 2. And the same for Ireland's goal probabilities with the appropriate parameters (n=9.12, p=7.6%). Then simply add the probabilities to get the probability of a Home, Draw or Away result.
The result is bad news for the Boys in Green. Binomial agrees with Poisson which also agrees with the Bookie! Both actually are giving Ireland less of a chance than the bookie gives.
Below is the full data used in the Binomial culcuations:
The Binomial Distribution describes another kind of random variable. It is a little more complicated than the Poisson but can also be applied to goals/football. It describes the number of positive outcomes in n trials given the probability of success p in each individual trial. In our case we will think of a shot as a trial, and the outcome is successful if the shot results in a goal. So to apply this we need to estimate the number of shots for a team per game (this will be n), and we will also need to estimate the probability that a shot results in a goal (this will be p). k is the number of goals.
So I looked back on 4 years data focussing on Scotland home matches and Ireland away matches. I picked 4 years because it is the natural cycle of international matches. Also, if I picked any shorter period then the number of matches would be too few to give a realistic long term average for our estimates of n and p. Poisson was a little easier as all we needed was the goals scored and goals conceded which is freely available on any results service. To estimate shots per game and shot success I needed to dig a little deeper and find total number of shots per game for and against. I had little success here but finally found on www.soccerway.com that they did have the data, but only for a limited number of matches in the last 4 years. I made no distinction between friendly and competitive matches.
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| Average shots per game (Scotland blue, Ireland Green - only counts home games for Scotland and away games for Ireland) |
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| Shot success (Scotland at home blue, Ireland away green - only counts home games for Scotland and away games for Ireland) |
So now to estimate our Binomial parameters based only on the above information.
n: This is easy. We will average out the number of shots per match. So Scotland average 10.83 shots per match in my sample. Ireland averaged 14.36 shots against. So it would be reasonable to assume that there will be about 12.59 (the midpoint) home shots in this match. By the same process I estimated that there would be about 9.12 shots for Ireland (away shots).
p: This is a little harder to figure out but still straight forward. Of Scotland last 65 shots that I have recorded off soccerway's data they scored 6. So by division the probability of a shot resulting in a goal is 6/56 = 0.0923. This is not the end of the story because its doesn't take into account Ireland's effectiveness as denying shots. So I calculate this too which is 15 goals conceded on the last 201 shots ie 0.0746 probability. Again average these out and it would be reasonable to expect that Scotland will score 8.3% of their shots against Ireland. Similarly if you apply the same process then Ireland will score 7.6% of their shots. We will take these percentages as the probabilities and use as p in the Binomial formula.
To calculate the Probability that Scotland score 0 goals we just fill in the parameters above (n=12.59, p=8.3%) for k=0. We do the same for k=1, 2. And the same for Ireland's goal probabilities with the appropriate parameters (n=9.12, p=7.6%). Then simply add the probabilities to get the probability of a Home, Draw or Away result.
The result is bad news for the Boys in Green. Binomial agrees with Poisson which also agrees with the Bookie! Both actually are giving Ireland less of a chance than the bookie gives.
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| Probabilities based on different methods |
Below is the full data used in the Binomial culcuations:
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