After the state pinball championship last month there was a second, side tournament. It was a sort-of marathon event in which I played sixteen games in short order. I won three of them and lost thirteen, a disheartening record. The question I can draw from this: was I hopelessly outclassed in the side tournament? Is it plausible that I could do so awfully?
The answer would be “of course not”. I was playing against, mostly, the same people who were in the state finals. (A few who didn’t qualify for the finals joined the side tournament.) In that I had done well enough, winning seven games in all out of fifteen played. It’s implausible that I got significantly worse at pinball between the main and the side tournament. But can I make a logically sound argument about this?
In full, probably not. It’s too hard. The question is, did I win way too few games compared to what I should have expected? But what should I have expected? I haven’t got any information on how likely it should have been that I’d win any of the games, especially not when I faced something like a dozen different opponents. (I played several opponents twice.)
But we can make a model. Suppose that I had a fifty percent chance of winning each match. This is a lie in detail. The model contains lies; all models do. The lies might let us learn something interesting. Some people there I could only beat with a stroke of luck on my side. Some people there I could fairly often expect to beat. If we pretend I had the same chance against everyone, though, we get something that we can model. It might tell us something about what really happened.
If I play 16 matches, and have a 50 percent chance of winning each of them, then I should expect to win eight matches. But there’s no reason I might not win seven instead, or nine. Might win six, or ten, without that being too implausible. It’s even possible I might not win a single match, or that I might win all sixteen matches. How likely?
This calls for a creature from the field of probability that we call the binomial distribution. It’s “binomial” because it’s about stuff for which there are exactly two possible outcomes. This fits. Each match I can win or I can lose. (If we tie, or if the match is interrupted, we replay it, so there’s not another case.) It’s a “distribution” because we describe, for a set of some number of attempted matches, how the possible outcomes are distributed. The outcomes are: I win none of them. I win exactly one of them. I win exactly two of them. And so on, all the way up to “I win exactly all but one of them” and “I win all of them”.
To answer the question of whether it’s plausible I should have done so badly I need to know more than just how likely it is I would win only three games. I need to also know the chance I’d have done worse. If I had won only two games, or only one, or none at all. Why?
Here I admit: I’m not sure I can give a compelling reason, at least not in English. I’ve been reworking it all week without being happy at the results. Let me try pieces.
One part is that as I put the question — is it plausible that I could do so awfully? — isn’t answered just by checking how likely it is I would win only three games out of sixteen. If that’s awful, then doing even worse must also be awful. I can’t rule out even-worse results from awfulness without losing a sense of what the word “awful” means. Fair enough, to answer that question. But I made up the question. Why did I make up that one? Why not just “is it plausible I’d get only three out of sixteen games”?
Habit, largely. Experience shows me that the probability of any particular result turns out to be implausibly low. It isn’t quite that case here; there’s only seventeen possible noticeably different outcomes of playing sixteen games. But there can be so many possible outcomes that even the most likely one isn’t.
Take an extreme case. (Extreme cases are often good ways to build an intuitive understanding of things.) Imagine I played 16,000 games, with a 50-50 chance of winning each one of them. It is most likely that I would win 8,000 of the games. But the probability of winning exactly 8,000 games is small: only about 0.6 percent. What’s going on there is that there’s almost the same chance of winning exactly 8,001 or 8,002 games. As the number of games increases the number of possible different outcomes increases. If there are 16,000 games there are 16,001 possible outcomes. It’s less likely that any of them will stand out. What saves our ability to predict the results of things is that the number of plausible outcomes increases more slowly. It’s plausible someone would win exactly three games out of sixteen. It’s impossible that someone would win exactly three thousand games out of sixteen thousand, even though that’s the same ratio of won games.
Card games offer another way to get comfortable with this idea. A bridge hand, for example, is thirteen cards drawn out of fifty-two. But the chance that you were dealt the hand you just got? Impossibly low. Should we conclude from this all bridge hands are hoaxes? No, but ask my mother sometime about the bridge class she took that one cruise. “Three of sixteen” is too particular; “at best three of sixteen” is a class I can study.
Unconvinced? I don’t blame you. I’m not sure I would be convinced of that, but I might allow the argument to continue. I hope you will. So here are the specifics. These are the chance of each count of wins, and the chance of having exactly that many wins, for sixteen matches:
|Contest||Probability the Higher Seed Wins||Information Content of this Outcome|
|#1 seed vs #16 seed||100%||0 bits|
|#2 seed vs #15 seed||96%||0.2423 bits|
|#3 seed vs #14 seed||85%||0.6098 bits|
|#4 seed vs #13 seed||79%||0.7415 bits|
|#5 seed vs #12 seed||67%||0.9149 bits|
|#6 seed vs #11 seed||67%||0.9149 bits|
|#7 seed vs #10 seed||60%||0.9710 bits|
|#8 seed vs #9 seed||47%||0.9974 bits|
So if the eight contests in a single region were all evenly matched, the information content of that region would be 8 bits. But there’s one sure and one nearly-sure game in there, and there’s only a couple games where the two teams are close to evenly matched. As a result, I make out the information content of a single region to be about 5.392 bits of information. Since there’s four regions, that means the first round of play — the first 32 games — have altogether about 21.567 bits of information.
A statistical analysis of the tournaments which I dug up last year indicated that in the last three rounds — the Elite Eight, Final Four, and championship game — the higher- and lower-seeded teams are equally likely to win, and therefore those games have an information content of 1 bit per game. The last three rounds therefore have 7 bits of information total.
Unfortunately, experimental data seems to fall short for the second round — 16 games, where the 32 winners in the first round play, producing the Sweet Sixteen teams — and the third round — 8 games, producing the Elite Eight. If someone’s done a study of how often the higher-seeded team wins I haven’t run across it.
There are six of these games in each of the four regions, for 24 games total. Presumably the higher-seeded is more likely than the lower-seeded to win, but I don’t know how much more probable it is the higher-seed will win. I can come up with some bounds: the 24 games total in the second and third rounds can’t have an information content less than 0 bits, since they’re not all foregone conclusions. The higher-ranked seed won’t win all the time. And they can’t have an information content of more than 24 bits, since that’s how much there would be if the games were perfectly even matches.
So, then: the first round carries about 21.567 bits of information. The second and third rounds carry between 0 and 24 bits. The fourth through sixth rounds (the sixth round is the championship game) carry seven bits. Overall, the 63 games of the tournament carry between 28.567 and 52.567 bits of information. I would expect that many of the second-round and most of the third-round games are pretty close to even matches, so I would expect the higher end of that range to be closer to the true information content.
Let me make the assumption that in this second and third round the higher-seed has roughly a chance of 75 percent of beating the lower seed. That’s a number taken pretty arbitrarily as one that sounds like a plausible but not excessive advantage the higher-seeded teams might have. (It happens it’s close to the average you get of the higher-seed beating the lower-seed in the first round of play, something that I took as confirming my intuition about a plausible advantage the higher seed has.) If, in the second and third rounds, the higher-seed wins 75 percent of the time and the lower-seed 25 percent, then the outcome of each game is about 0.8113 bits of information. Since there are 24 games total in the second and third rounds, that suggests the second and third rounds carry about 19.471 bits of information.
Taking all these numbers, though — the first round with its something like 21.567 bits of information; the second and third rounds with something like 19.471 bits; the fourth through sixth rounds with 7 bits — the conclusion is that the win/loss results of the entire 63-game tournament are about 48 bits of information. It’s a bit higher the more unpredictable the games involving the final 32 and the Sweet 16 are; it’s a bit lower the more foregone those conclusions are. But 48 bits sounds like a plausible enough answer to me.
Yes, I can hear people snarking, “not even the tiniest bit”. These are people who think calling all athletic contests “sportsball” is still a fresh and witty insult. No matter; what I mean to talk about applies to anything where there are multiple possible outcomes. If you would rather talk about how interesting the results of some elections are, or whether the stock market rises or falls, whether your preferred web browser gains or loses market share, whatever, read it as that instead. The work is all the same.
To talk about quantifying how interesting the outcome of a game (election, trading day, whatever) means we have to think about what “interesting” qualitatively means. A sure thing, a result that’s bound to happen, is not at all interesting, since we know going in that it’s the result. A result that’s nearly sure but not guaranteed is at least a bit interesting, since after all, it might not happen. An extremely unlikely result would be extremely interesting, if it could happen.
What We Talk About When We Talk About How Interesting What We’re Talking About Is | nebusresearch, Joseph Nebus, elkement, and 2 others are discussing. Toggle Comments
The Prior Probability blog points out an interesting graph, showing the most common scores in basketball teams, based on the final scores of every NBA game. It’s actually got three sets of data there, one for all basketball games, one for games this decade, and one for basketball games of the 1950s. Unsurprisingly there’s many more results for this decade — the seasons are longer, and there are thirty teams in the league today, as opposed to eight or nine in 1954. (The Baltimore Bullets played fourteen games before folding, and the games were expunged from the record. The league dropped from eleven teams in 1950 to eight for 1954-1959.)
I’m fascinated by this just as a depiction of probability distributions: any team can, in principle, reach most any non-negative score in a game, but it’s most likely to be around 102. Surely there’s a maximum possible score, based on the fact a team has to get the ball and get into position before it can score; I’m a little curious what that would be.
Prior Probability itself links to another blog which reviews the distribution of scores for other major sports, and the interesting result of what the most common basketball score has been, per decade. It’s increased from the 1940s and 1950s, but it’s considerably down from the 1960s.
You can see the most common scores in such sports as basketball, football, and baseball in Philip Bump’s fun Wonkblog post here. Mr Bump writes: “Each sport follows a rough bell curve … Teams that regularly fall on the left side of that curve do poorly. Teams that land on the right side do well.” Read more about Gaussian distributions here.
The “God Plays Dice” blog has a nice little baseball-themed post built on the coincidence that a number of the teams in the postseason this year are from the same or at least neighboring markets — two from Los Angeles, a pair from San Francisco and Oakland, and another pair from Washington and Baltimore. It can’t be likely that this should happen much, but, how unlikely is it? Michael Lugo works it out in what’s probably the easiest way to do it.
The Major League Baseball postseason is starting just as I write this.
From the National League, we have Washington, St. Louis, Pittsburgh, Los Angeles, and San Francisco.
From the American League, we have Baltimore, Kansas City, Detroit, Los Angeles (Anaheim), and Oakland.
These match up pretty well geographically, and this hasn’t gone unnoticed: see for example the New York Times blog post “the 2014 MLB playoffs have a neighborly feel” (apologies for not providing a link; I’m out of NYT views for the month, and I saw this back when I wasn’t); a couple mathematically inclined Facebook friends of mine have mentioned it as well.
In particular there are three pairs of “same-market” teams in here: Washington/Baltimore, Los Angeles/Los Angeles, San Francisco/Oakland. How likely is that?
(People have pointed out St. Louis/Kansas City as being both in Missouri, but that’s a bit more of a judgment call, and St. Louis…
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