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 1/8th 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.
11 thoughts on “The Playoffs Are a Crapshoot, Part 2: Basic Probability Math”
JC’d and this seems to fit nicely with the post:
Apologies if this steps on the toes of future posts, but in looking at the last 10 playoff series, the Braves teams had an average regular season winning percentage of .582 compared to an average winning percentage of their opponents of .562. Couple that with the fact that ATL had homefield in 7 of the 10 series and the pre-series odds on average had to be a little better than 50/50 I’d say.
Yes, Dusty. Winning percentage isn’t the best measure of quality because of unbalanced schedules, but it’s pretty close. So you might expect the Braves to have about a 52-53 percent chance to win any particular series. (It has to be considerably lower than their 58 percent chance of winning a game against an average team, given the 56 percent winning rate of the opponents.) So with three series to win, the probability of winning the World Series would be about .525^3, or about 14.4%…. about one win in 7 attempts rather than the one win in 8 attempts if things were completely random. But the wild card years have lower probabilities (about half) because of the extra round. In 10 years of this, they should have won about 1, not 0. That’s the back-of-the-envelope way to see where we’re going here.
I think that there ARE things that are important to 162 game success that aren’t as much to “best 3 of 5” or “best 4 of 7” success.
Spots 4 through 8 of a bullpen.
Starters 4, 5, and 6.
Second or back up catcher (if not a platoon).
System wide depth.
Each of these 4 things helps a team become a division champion, but has little effect on post season.
There are also things that either have no effect or negative effect in a 162 game season that have positive effect in a short series.
bunting / situational hitting.
ability to steal bases, successfully.
Limited platoon hitting disadvantages.
Even as someone who didn’t enjoy statistics class in college, this is still a very interesting topic, JonathanF.
I thought my browsing issues were over, but I still have to remember to refresh the page.
Happy new year to all, in case I’m too out of it to type later tonight. May the new year be better than the previous ones, for us as brave fans and personally as well.
Great stuff, JonathanF. Cliff, I’m with you entirely @3, but I’m not convinced @4. Bunting and stealing are overrated, but they can be effective late in a low scoring, close game. Is it really the case that there are more low scoring and close games in a short postseason series than in the regular season? There may very well be, but I would not assume that without evidence.
Carl, we are planning a migration to a new host in the next few days that will finally get things more in hand. We finally got access to the backend of the site this week, making some improvements possible going forward.
I teach mathematics at Clemson — never expected to see mathematical argumentation here. Bravo! :)
New thread! We will return to JF’s series tomorrow! Happy New Year’s, Braves Journal!
Thank you, Jonathan.
Good luck on the 13th, sdp. If the world survives until then, the probability is 100 % that the Tigers win.
There?s definately a great deal to know about
this topic. I like all of the points
you made. https://www.myozen.ca