Irelands EURO 2016 playoff: Mathematically speaking

There were plenty of twists and turns to see which teams would make the playoffs, and which ones would be seeded for the draw. All was finally decided with an obscure goal in Riga scored by Kazak star Islambek Kuat. This goal in combination with a 89th minute strike in Istanbul changed everything literally in the last minute.

But now we are where we are and Ireland face one of the below seeded teams.

FoG blog decided to do a little analysis to check Ireland's probability of winning the playoff against each team.

Using Mathematical models we previously predicted the winner of the League of Ireland decider clash at the end of last season. We also predicted the Scottish win against Ireland last November.

We used the Poisson model again and the results are below.

There is not a huge difference in the teams - ranging from a 45% chance for Ireland to qualify if drawn against Ukraine to a 59% chance against Hungary. This is to be expected with all teams of roughly the same quality - we've all arrived in this situation in the same manner. We've all failed to qualify automatically.

On average the chance of qualification is 50.34%. 

The verdict

Right now it looks like a flip of a coin whether Ireland qualify. Drawing Sweden or Hungary will slightly increase our chance. Drawing Bosnia or Ukraine will slightly reduce our chance. Hungary gives us the most likely chance. Ukraine gives us the least likely chance.

The Maths

Okay I went easy on the Maths behind the prediction so far for fear that I would lose you. Sticking it in here for those interested but feel free to stop reading here.

The Poisson formula is very easy to apply as you only need one single parameter - some estimate on the average or expected number of goals. It is also very accurate for describing any discrete arrival random variable, for example number of cars arriving at a red light per minute, number of claims arriving at an insurance company per day, the number of goals per 90 minutes.

The formula itself looks complicated, but there is only one unknown in the formula and its quite easy to take a long run average of goals scored/conceded at home or away for each team to estimate the unknown parameter.

So to translate that to English and apply to our example: 

"The probability that Ireland score k goals = average goals ^ k time euler's number ^ - average goals all divided by k factorial."

So that's not really English but you get the idea - k, e and k! are all given, lambda (the funny symbol) is all that we need to figure out. This is the expected goals and we can estimate based on historic results.

So take Ireland versus Hungary. I see that Ireland have score on average 1.86 goals per match at home, and I see that Hungary have conceded an average of 2.1 goals away from home. So it is reasonable to expect Ireland to score an average of 1.98 goals (the midpoint of Ireland expectation to score and Hungary's expectation to concede) at the Aviva against teams like Hungary. Repeat this to estimate the number of goals conceded by Ireland at the Aviva, scored in Budapest and conceded in Budapest.

Its a little tedious but this then gives us an expected score for the aggregate score of each match. We then can work out the probability of each possible score based on these parameters.

For example:

So the probability that Ireland beat Hungary 2-1 on aggregate can be calculated

P[Ireland=2] = 2.93 ^ 2 * 2.71828 ^ -2.93 / 2! = .229

P[Hungary=1] = 2.41 ^ 1 * 2.71828 ^ -2.41 / 1! = .216

(Tip: copy and paste the formulae into google to check I'm not kidding you).

So to get the probability that Ireland score 2 and Hungary score 1 you multiply both probabilities, so 0.229 * 0.216 = 0.049464. That gives just less than a 5% chance of happening. If you continued to calculate the probability of all the possible scores and summed the probabilities for wins draws and loses then you would come to the probabilities that I have posted here above. In the case of a draw over two legs I have assumed a 50-50 chance of qualifying.

Would seeding have made a difference? For arguments sake imagine if Cyprus had dumped out Bosnia and we had drawn with Poland then we would have been seeded and facing a slightly better chance of qualifying based on probabilities.