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Wednesday, September 14, 2011

Magic Numbers, Magic Percentages, and Magic Probabilities

The logic here could be applied to any sport, but it is most applicable in Major League Baseball (and it is in this context that I had the idea). I love magic numbers in baseball*. As a Rangers fan, I usually start tracking their magic number before the All-Star break. There is something so definitive about a magic number. But magic numbers really tell you very little about how likely it is that a team will win a division.

So, to extend the magic number, I've added magic percentages and magic probabilities. The magic percentage is the magic number divided by the number of games remaining for both teams. For example, if a team has 10 games left to play and a they have a 3 game lead on a team that also has 10 games left, the leading team will need 40% ((10-3+1=8)/(10+10=20)=.4) of those 20 games to have a favorable outcome - they win or the second place team loses - to clinch. The second place team in this situation has a magic number of 14 and a magic percentage of 70% - they need 70% of games to have a favorable outcome to clinch. The two magic percentages do not add to 100% because there is the additional possibility that the teams tie at the end of the season, which would not satisfy the requirements of the magic percentage for either team.

Calculating the magic probability is a bit more complicated, but there are plenty of tools to help. Here's the logic: if we know a team needs 40% favorable outcomes, what is the probability that they will get that and clinch the division? We can calculate this, using some simplifying assumptions, by drawing on the binomial cumulative distribution function.We are going to assume that the team has a 50% chance of getting a favorable outcome in each game (it will actually be a bit higher for their games, but a bit lower for their opponent's games, so it averages out, more or less). With that assumption, we can go to the calculator below. n is the number of games remaining for both teams (20 from the example above), p is .5, Prob. X is should be set to "more than", and the next blank takes the magic number minus 1. Hit compute. So, for the example, we find that a team with a three game lead with 10 games left to play has a 87% chance of winning the division (without taking into account schedule or other idiosyncrasies). If both teams win the next 5 games, the magic number will have shrunk to 3, the total remaining games to 10, and the magic probability will have risen to 95%.

*I'm going to assume above that readers are familiar with the concept of magic numbers, but for those that aren't, a team's magic number is the number of favorable outcomes-wins for Team A or losses by the team in their division with the best record (not counting the Team A, of course)-they need to clinch the best record in the division. It's calculated as Games Remaining-Lead in the Loss Column+1. If the team is not in first place, Lead in the Loss Column will be a negative number. So, if a team has 10 games left and a 2 game lead on the second place team, their magic number is 9. They can clinch the division by winning 9 games, by the 2nd place team losing 9 game, or some combination thereof.

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