Saw the USA beat Algeria today, so naturally I’m a fútbol expert, though the refs make me miss the NBA. Casualty Actuarial Society passes along a study by pension actuary Mitch Wiener of probabilities of making the knockout round; here’s the link. It’s a fairly clinical look at Bayesian estimation – the probability of success given a certain situation.
Here is the probability of advancing given a certain number of points:
First, the basics: You get three points for a win, no points for a loss and one point for a tie. Each group has four teams that play round-robin. At the end, the two teams with the most points advance.
As for this chart, probably the easiest way to understand it is by looking at the ‘Number’ column on the left and the ‘Total’ column to the right. If you score ‘9’ points in the first round (‘9’ in the number column), the probability of you advancing is 100% (taken from the ‘Total’ column). If you score seven points, the probability of advancing is still 100%. (It’s impossible to score eight points, by the way.)
If you score six points, the probability of advancing is near certain – 97.3%. But there’s one circumstance – three teams wins two games and one wins no games. Then one of the six-point winners will miss out. And so on.
Now this analysis has three big flaws. First, it assumes that each team has a 50-50 chance of winning each game. And I’m here to tell you North Korea could play Portugal 99 times and earn a tie at most once, if there was a blackout in the first minute. (The actual result was 7-0.)
The second problem is a bit tougher to explain. The analysis assumes that all games are independent – that is, the result of a team’s third game is independent of that team’s second game and also independent of every other game played or being played. And that’s certainly not the case.
Suppose Team A is clearly better than the other teams in its group and that, further, Teams B, C and D are so evenly matched that they could only manage draws against each other. After two rounds, Team A would have six points and would have clinched advancing, since Teams B, C and D would have only one point each. At this point, Team A could lose intentionally to determine who could advance.
Don’t laugh. West Germany and Austria rigged a similar situation so blatantly (against Algeria) that the game has its own Wikipedia entry.
Which leads to the third failing. As a sports fan, I don’t care what the probability of advancing is after a team has played all of its games. Once the games are played, I know exactly who advances. What I really want to know is what a team’s chances are after one game is played or after two games are played.
For that, you’re much better off hanging out at Nate Silver’s site, fivethirtyeight.com.