There are three games left in the 2012 MLS season: two conference final games, then one game for the title.
The conference final match-ups follow a two legged format - one game is played at each team's home field and the winner is whoever scores the most goals across both games. In the case of a tie, OT, then penalty kicks, is played in the second game. This gives a small advantage to the team playing at home in the second game (the higher seeded team).
One game in each conference final has been played. LA leads Seattle 3-0 with the remaining game to be played in Seattle, and Houston lead DC 3-1 with the remaining game to be played in DC.
I ran a Monte Carlo simulation of the remaining three games and was surprised to learn that DC United are not in as weak of a position to win the MLS Cup as their current deficit implies. See below:
Despite being down two goals going into today's second leg, DC has nearly as good a chance to win the MLS Cup as Houston, for the following reasons:
- Home field advantage in Major League Soccer is quite strong.
- DC has been excellent at home: they've allowed just 1.0 goals/game while scoring 2.2 goals/game.
- Houston has been weak on the road, allowing 1.7 goals/game and scoring 1.0 goals/game
- Today's second leg game is played in DC.
- If DC makes the championship, that game will be in DC. If Houston makes it, the championship would either be in LA or Seattle.
Actual Math (very short though) below.
I assume goals are Poisson distributed with an average calculated as follows, using Houston and DC as an example:
- Houston (Goals scored by Houston on the road + Goals allowed by DC at home) / 2 = 1.0
- DC (Goals scored by DC at home + Goals allowed by Houston on the road) / 2 = 1.94
The Poisson means that goals are scored at random times, but on average they happen 1.94 times/game for DC. This translates into 0 goals some of the time, 1 goal some of the time, 2 goals, 3 goals, etc. with decreasing frequency.
I simulate OT (if needed) using the same method with the same Poisson averages divided by 3. If the match goes to PKs, I assign 50% win probability to each time. The assumptions inherent with the methodology are:
- Goals scored this season (at home and on the road) are a good predictor of future goals.
- Goals are Poisson distributed.
- Each team has a 50/50 chance when the game goes to PKs.