When I was 8 or 9 years old, a gifted classmate named Daniel taught me how to draw a cartoon face of a man with a mustache. This was no easy trick because then, like now, I had absolutely no artistic talents whatsoever. None. Whenever I have been placed in the uncomfortable decision of having to draw something -- for the purposes of a game or based on a daughter's request or whatever -- the results have been either comic or tragic, depending on which one more poorly suited the situation. I could draw a smiley face right now and you would say: "Um, is that supposed to be a vase? A house? John Cusack?"
But I can still draw that face of the man and his mustache. And I do. All the time. It looks more like man meets walrus, yes, but it's the only thing I know how to draw and this is my mode: When I learn how to do something, no matter how weak it may be, I tend to stick with it.
So, some years ago, a friend taught me how to figure out the basic odds of a hitting streak. This too is a limited mathematical trick for many reasons, especially how I do it. But it's a fun trick, especially when a guy like Dan Uggla suddenly goes on a 33-game hitting streak.
First: You need to figure out percentage chance that a player will get a hit in any given game. This can be figured out by multiplying the chances that the hitter will not get a hit by each other. For instance, let's take a pure .300 hitter -- that is a player whose true batting average is .300. This assumes that a player can have a true batting average and that we can figure it out and that every pitcher offers the same odds of getting a hit, just three of the reasons this is such a limited trick. But, hey, we're just having fun.
So there's a 70% chance he won't get a hit in his first at-bat. And there's a 49% chance he won't get a hit in his first two at-bats. And so on. It's better once you subtract and come up with chances he WILL get a hit.*
*I have updated this so that it now uses AT-BATS rather than PLATE APPEARANCES. That was a fundamental mistake in the original version.
Chances he will get a hit in 1 AB: 30%
Chances he will get a hit in 2 AB: 51%
Chances he will get a hit in 3 AB: 66%
Chances he will get a hit in 4 AB: 76%
Chances he will get a hit in 5 AB: 83%
And so on. Obviously these are simply raw numbers. How well do they match up with reality? Not bad. Since 1950 there have been three players with 3,000 plate appearances who hit .300 on the nose -- Robbie Alomar, Pedro Guerrero and John Kruk. And in games where they got at least three at-bats, here's how often they got at least one hit:
Robbie Alomar: 76.3%
Pedro Guerrero: 74.3%
John Kruk: 73.1%
So, yes, it fluctuates … I suspect the biggest fluctuation comes from how many at bats a player gets per game. And, of course, people are different. Games are different. Situations are different. Some people walk more often, and a walk doesn't count as either a hit or an at-bat. But, please remember, this is all just a trick.
OK. Between 1936 and 1941, Joe DiMaggio hit .345. Punching that into the system:
Chances he will get a hit in 1 AB: 35%
Chances he will get a hit in 2 AB: 57%
Chances he will get a hit in 3 AB: 72%
Chances he will get a hit in 4 AB: 82%
Chances he will get a hit in 5 AB: 89%
So that should give you an idea of how often a .345 hitter should get a hit in a game. The DiMaggio reality? Well, he got 3 or more plate appearances in 812 games in those six years. He got a hit in 663 of them -- or 81.7% of the time. So that more or less matches up with the stats.
So, that's the first step: Figure out how often a player should get a hit in a game. To figure out the percentage for Uggla 33-game hitting streak, we start with what the chances are that he will get a hit in a game. He is a lifetime .259 hitter. He hit .287 last year. When the streak began, he was hitting .173. So it's not an easy thing to figure. Taking Uggla's lifetime batting average:
Chances he will get a hit in 1 AB: 26%
Chances he will get a hit in 2 AB: 45%
Chances he will get a hit in 3 AB: 59%
Chances he will get a hit in 4 AB: 70%
Chances he will get a hit in 5 AB: 78%
OK, so as you probably guessed, he's not quite as good at this hit-making business as DiMaggio. His career numbers more or less match up with the odds. Before the streak, he had gotten a hit in roughly 63.5% of his games. Uggla has walked a fair amount in his career, which is probably why he skews closer to three at-bats than four.
Once you get the true chances of a batter getting a hit, you can multiply that percentage by itself again and again to get your basic hitting streak chances. I find it easiest when thinking of a coin flip. The odds of you picking up a quarter right now and getting heads is 50%. The odds of doing it two times in a row is 25%. Three times in a row is 12.5%. And so on.
But the math is tricky. Because while the overall odds can get astronomical -- the odds of a player with the hitting talents of Joe DiMaggio hitting in 56 straight games starting on any given day are about 82,000 to 1, and 57 straight are more than 100,000 to 1 -- but the daily odds stay exactly the same. That is to say that when you flip a coin, the chances of flipping two heads in a row are 25%. But once you flip a heads, your chances of flipping another one jumps right back to 50%. DiMaggio began every game with roughly the same odds of getting a hit whether it was Game 2 or Game 56 (obviously taking into account the effects of pressure, pitching, weather and so on).
So, no matter how big the Uggla numbers get -- and they're already absurdly huge -- he's still more likely to get a hit today than not get a hit.
Uggla, because he has only gotten hits in 63.5% of his games, is exponentially less likely to go on a lengthy hitting streak than DiMaggio. You've heard the story about the chess player who, when offered anything he wanted, said he wanted a single grain of rice doubled for every square on a chess board. Well, anyway, that's how I heard it. So, one grain of rice, then two, then four, then eight, then 16, then 32, then 64, then 128, doubling until all 64 squares on the board were accounted for. Even knowing the answer, it is shocking. This simple exercise ends up with more than 18 quintillion grains of rice.
You wouldn't double Uggla's chances of not getting a hit each game … but close enough. The pure percentages suggest that the odds are 4-to-1 against him going on a THREE game hitting streak starting today. The odds of Dan Uggla on July 5, 2011 beginning a 33-game hitting streak were more than three million to one against. The odds of it being a 34-game hitting streak? Almost five million to one. Then, it jumps to 7.8 million to one, 12 million to 1 and so on.
The odds of Dan Uggla on July 4, 2011 beginning a 56-game hitting streak? Right. That's about 108 billion to 1, give or take a few million.
But, again, the math is tricky. Now that he's here are 33, the odds of him getting a hit in his next 24 games are much more manageable … they are about 50,000 to 1 against. And that's child's play for the new Dan Uggla.
Obviously, I'm mangling this to death. There are so many other factors, including the fact that a season is continuous and a hitting streak can start at any time and so on. But since we've already gone this far, we might as well go on: Many people have asked: What's statistically less likely -- DiMaggio's 56-game hitting streak or Ted Williams' 84-game on-base streak?
Williams reached base 84 straight games in 1949. He also reached base every game from July 20, 1941 to the end of the season. That guy was pretty good at getting on base.
So to figure the 84-game on-base streak … one more time, we need to figure out the odds of Williams reaching base in a game. Well, I'm going to tell you something amazing. Between 1939 and 1958, that's 20 years, Ted Williams played in 1,980 games when he got at least three plate appearances. Almost 2,000 games. Do you know how many times he did not reach base? Go ahead. Take a guess.
Wrong. The answer: 139.
Yeah. That's 139. The man reached base in NINETY THREE PERCENT of the games when he got at least three plate appearances. The chances of Ted Williams reaching base in a game was higher than Rick Barry's free-throw percentage.
Amazingly, that 93% number is about what you would expect from someone who reaches base almost 50% of the time.
So, what are the odds of someone who gets on base in 93% of his games going on an 84-game on-base streak? Well, it's the same odds as a 93% free-throw shooter making 84 straight free throws. It's high, of course, but not NEARLY as high as DiMaggio's hitting streak odds.
You will remember:
DiMaggio odds of 56 straight hits was about 82,000 to 1.
Williams odds of 84 straight games reaching base: About 406 to 1.
Quite a difference there. But, this doesn't necessarily mean DiMaggio's streak is more impressive. It really means that Ted Williams was incredibly awesome at getting on base. The odds of DiMaggio going on that 84-game on-base streak -- as great as he was -- is more than 23,000 to 1.
My apologies to math teachers everywhere.
Sunday, August 14, 2011
