The Playoffs Are a Crapshoot, Part 2: Basic Probability Math

If you missed the first piece in “Playoffs are a Crapshoot”, it provided a basic overview of what will be discussed and you can find it here.

There are 10 teams that make the playoffs every year, six division winners and 4 wild card teams.  Suppose that the probability of each of them winning the World Series is 10%.  If that were true, that would be the ultimate crapshoot.  There would be no difference (in probability of winning the World Series) between playing the postseason out and picking one ball out of a well-shaken urn containing ten balls.  The ratings would be higher for the baseball games (I hope!) even though there are a lot more commercials, but at least for nine of the teams, the misery would be over pretty quickly.

Actually, because of the Wild Card game, where two teams get eliminated after one game, that’s a little too flat.  The real ultimate crapshoot would be where the wild card teams had a 6.25% chance of winning and the division winners have a 12.5% chance of winning.  The playoffs would consist of two coin flips and then the selection of one team out of eight from the urn…  sponsored by Urns ‘R Us, at least until they are disintermediated by urns.com. 

On the other hand, suppose we could rank the teams in advance and the top team had over a 50% chance of winning the Series.  (The 50% here is arbitrary, but if the probability were that high, I think it would be fair to say that the results were far from random.) 

So which is it?  It’s quite simple to show that reality has to be a lot closer to random than to a team having a 50% chance of winning.  A team needs to win either 11 or 12 games to be World Series champs.   The number of games that need to be played is somewhere between 11 and 20.  Let p be the probability of winning any particular game.  Then, to win a World Series, a team must (a) win 3 games before it loses 3; and then, twice, (b) win 4 games before it loses 3.  If the team is a wild card team, it must win the wild card game as well.

Let’s say a team has a 65% chance of winning every game it plays.  This is ridiculously high.  A team that wins 65% of its games in the regular season, most of which are played against teams much worse than the teams in the playoffs, wins 105 games.  How often will such a team win the World Series?  If they won their division, the chance of winning a divisional series (3 out of 5) is 76% and their chance of winning the LCS and WS are each 80%.  (The note at the bottom derives these probabilities.)  So, assuming they aren’t a wild card, their chance of winning the World series is 0.76 x 0.8 x 0.8 = 49%, just under 50 percent. If they were the wild card team, their chance of winning falls by 35% (since they have to win the wild card game as well) to 32%.  Our only data: two teams in the modern era have won 120 games. One won the World Series.  The other one lost in the LCS round.

In order to have a 90 percent chance of winning the World Series, a team would have to have an 81 percent chance to win every game.  While teams have occasionally played that well for a month, there is no way that performance represents skill alone – skill differentials between teams are not high enough to make that happen.

On the other hand, consider a pure crapshoot.  A team with a 50% chance of winning every game has a 50% chance of winning any series it plays.  This translates into a 12.5% chance of winning the World Series as a division winner and a 6.25% chance of winning as a Wild Card team.  This averages to 10 percent across the ten teams and is dead flat across the eight teams playing in the divisional series, simply creating a 1/8^th chance for all 8 teams.

So now we can now measure the crapshootiness of the playoffs by simply seeing how far above 12.5% the probability of the most likely team to win the World Series (I call this pbest) is.  It can’t be any lower than 12.5% and it is very unlikely to be anywhere near 50%.  I will define here a Crapshootiness Index as 1-(pbest-0.125)/0.875, which would measure 1 in the pure crapshoot case and 0 if the best team had a 100% chance of winning.   How to measure pbest, the probability of the best team in the playoffs, is parts 3 and 4 of the series.

I note that the probability of the team with the best chance isn’t the only way to measure crapshootiness.  We might measure it, for example as

where the first term sums the absolute deviation over the 4 wild card teams to win the World Series and the second term sums the absolute deviation over the six division winners.  That’s really not much harder to calculate, but it’s a little harder to put in context rather than the 0-1 measure I’m proposing.  The other reason is that it is sensitive to bad teams in the playoffs having little chance to win.  I don’t think that’s what we mean when we say “the playoffs are a crapshoot.”  What we mean is that good teams don’t have as good a chance as it seems.  The addition of bad teams ought to reduce the crapshootiness of the playoffs, and in my measure, they do.

Note on derivation of probabilities: Let the probability of a game win be p.  In a 3 out of 5 series, there is one way to win in 3 games, 3 ways to win in 4 games (the loss could come in either game 1, 2 or 3) and 6 ways to win in 5 games (the two losses coming games 1-2,1-3,1-4,2-3,2-4, or 3-4).  Thus, the aggregate probability is p³ + 3p³(1-p) + 6p³(1-p)².  Similarly, for 4 games out of 7, there is one way to win a 4 game series, 4 ways to win a 5 game series, 10 ways to win a 6 game series and 20 ways to win a 7 game series, creating the aggregate probability p⁴ + 4p⁴(1-p) + 10p⁴(1-p)² + 20p⁴(1-p)³.

Thanks for reading “Playoffs are a Crapshoot, Part 2”. If you enjoyed this piece, check out this other piece by Jonathan F, which is a gloriously written piece on Mike Soroka.