The first law of large numbers, as everyone knows, is that as n gets high, the sum of n trials of a random variable (e.g. number of heads in n tosses) will asymptotically approach its expected value (e.g. n/2), with the expected average deviation (e.g. difference from n/2 divided by n) approaching 0. To be precise, the standard deviation is proportional to the square root of n. But there’s another, entirely different, truth which I call the “second law of large numbers,” which goes like this: given a large number of possibilities, a lot of improbable things will happen. In this post, I will talk about this law applied to sports, but it works equally well in real life.
Everyone knows that when a pitcher throws a no-hitter, he has great stuff, right? Or that basketball players get hot during games — Kobe Bryant went 28 for 46 in scoring 81 points, he was on fire tonight! Except it turns out that you don’t need those explanations — there are a ton of opportunities for a pitcher to throw a no-hitter, and a ton of opportunities for a player to go 28-for-46 (which is actually not that impressive — it’s only about +2 standard deviations for a player shooting 45% from the field), and just generally a ton of opportunities for things to happen. Most of the time, when something seemingly improbable happens, it’s random. This doesn’t mean it shouldn’t be analyzed — after all, some of these are perhaps not random, and the tricky thing about randomness and probability is that you can never tell random from nonrandom on an individual trial. But it does mean that you should take all such analysis with a grain of salt.
Recently, this has especially been hammered home by simbase. simbase is a simulated baseball league, currently enjoying its third incarnation. The thing about simbase is that we know that there is no such thing as “being hot” or “being clutch” — I wrote the engine myself, and there is no hanky panky going on. Players have fixed abilities, and while there are nonlinear batter-pitcher interactions in this incarnation, their effect is likely minimal. And yet, even in simbase with none of this pop psychology or variation in true talent, seemingly improbable things happen all the time. We’ve played about as many games so far as in a full baseball season in one real major league (the AL, since we have a DH), with roughly the same level of overall offense, and tons of things have happened.
There have been three no-hitters, including one in the playoffs, and including one by a mediocre-at-best pitcher who wasn’t even good enough to keep a starting spot. We had a player retire the first 25 batters in a playoff game only to give up a solo home run and lose 1-0. We had a 22-inning game. We had a player hit two home runs in each of three consecutive games: a player who hit four home runs total in his other 31 games that season, and another player hit 10 home runs in the second half of a (50-game) season after being possibly the worst player in the league in the first half.
We had a 20-pitch at-bat leading off a game; we had a team catch fire in the second half with no personnel changes, going 20-4 over its last 24 games after having gone 11-15 over its first 26, only to lose out to an equally hot team (literally: two teams in a pennant race both went 20-4 to close the season) by a single game. We had a seemingly unstoppable team go 38-12 in the regular season (and something like 35-9 before resting for the playoffs), then lose the first three games of the playoffs (and eventually that first-round series in six games). We had the two-time defending champions, who had a 37-13 record and had won eleven in a row during one particularly hot streak in the regular season, lose in the first round to a team that had never made the playoffs before and that was 1-5 against the champions in the regular season over those first three years.
The point, of course, is that all of these very unlikely things happened because of the second law of large numbers. There is absolutely no fluctuation within a season in a player’s true talent — things just happen because there are so many possible unlikely things. If anyone still thinks that every time a streak happens a player or team is hot or cold, I suggest you play some sort of automaton-based simulation game — it will quickly disabuse you of that notion.
