# Premier League Probabilities – Backgrounder

In 2009 I taught a course at the U of A’s School of Business on simulation – the application of quantitative techniques to answer business questions where there is uncertainty or risk. For example, if you run a retail store and are trying to manage inventory, you need to have an idea of your demand so you can decide how much to stock. But demand is unknown – so, by using random variables based on previous demand, then repeating the analysis many times over, you can understand the effects of changes to your bottom line. That’s the power of simulation.

One of the things we did in the course to keep interest high and demonstrate the flexibility of simulation was to predict the outcome of the NHL regular season, an idea first brought to the class by Armann Ingolfsson. Starting at the All-Star Break in January, we used Monte Carlo techniques to simulate the outcome of all remaining games in the NHL. A win percentage was derived for each team based on their performance. For each game that remained on the schedule, the relative performances of each team would be used to determine the probability of a win, a tie, and an overtime win. If two evenly-matched teams played, the outcome was essentially a coin toss. If a strong team played a weak one, there would be a greater chance for the favourite to win but there would always be a chance for the underdog. After simulating the outcome of all remaining games, you can figure out who finished where, who made the playoffs, and even what the first-round matchups would be. When you repeat this process, say 10,000 times, you can determine the probability of a team’s making the playoffs. When I last taught the course, in 2009, Edmonton’s playoff hopes came down to the last 10 or so games (ultimately they missed the cut). The fortunes of the Copper & Blue have changed, of course, and as I gear up to teach the course again in 2014, I wonder whether it’s even worth the effort to track Edmonton’s playoff opportunities!

In 2011, as the 2010-2011 Premier League began its last 10 games, some big stories began shaping up – Man United or Arsenal for the League title? Who would get the fourth Champions League spot? Would all teams starting with “W” face relegation? – and it occurred to me that those same techniques that went into estimating the Oilers’s playoff chances could be used to put some numbers to the debates. I set up a simulation model that was basically identical to our hockey version.

Coming soon: a more detailed description of how the simulation works. For the moment, some bullet points:

• Take the current standings in the Premier League, making sure you have both home and away statistics (Win/Draw/Loss)
• For each team calculate a win percentage (number of wins divided by total number of games) and number of draws (same method). It follows that loss percentage is 1-(win percentage + draw percentage) since the three options fully define the outcomes, or in technical terms  “mutually exclusive and collectively exhaustive.” We establish home and away percentages separately since home field advantage is critically important in soccer matches.
• For each remaining match in the season, calculate the probability of the home team winning as the average of the home team’s “home win” percentage and the away team’s “away loss” percentage.
• Calculate the probability of a draw as the average of the home team’s “home draw” percentage and the away team’s “away draw” percentage.
• Draw a random variable to simulate the outcome of the game using the probabilities calculated for that game. Assign an outcome, and repeat for all remaining games.
• Add up the simulated stats to determine the final point total for each team. Since the Premier Leage uses goal differential to settle ties, I assign a +1.5 to goal differential for every win and -1.5 for a loss (based on the average goal differential per game).
• Pick and track the results of interest  – who will win the League, who is relegated, how many points are needed to win or stay up, etc.
• Repeat the simulation many times to develop probability distributions for all outcomes.