I’m not much of a baseball fan, but we’re edging our way toward football season, so I flipped to ESPN radio a couple of days ago, in time to hear Mike and Mike discussing Jim Thome’s 600th home run. They were questioning how much meaning we should attach to home run records any more, given how many players were using steroids and other performance-enhancing drugs. In support of the record being a big deal, they played a clip of ESPN analyst Bobby Valentine pointing out that even with the steroid-inflated batting numbers, not that many guys are making a serious run at this particular milestone. This included the line:
Only 7 guys have done this [Thome is the 8th], but 17,000 have tried.
Kate was in the car with me, and we both had the same thought: “That’s a weirdly specific number.”
Now, baseball is sufficiently stats-obsessed that there probably is a complete tally of every player who ever played somewhere, and ESPN employs a bunch of researchers, so 17,000 might be a real figure. Then again, it’s Bobby Valentine, and it wouldn’t be the first time he just made shit up to have something to say.
I don’t care enough to look for the real figure, if it exists, but with a little bit of thought, we can at least see if Valentine’s figure is reasonable, Fermi problem style.
So, how many players have there ever been in major league baseball? To estimate this, we need to start with how many players are in the league now, and then work backwards. We’re only looking for an order of magnitude kind of number, here, so these will be rough figures.
There are around 30 teams in the league, and each team has something like 30 guys on the roster at any given time. Multiplying those together would give you 900 current players, but there are always a few guys who get called up partway through, so let’s call it 1000, to have a nice round number.
So there are 1000 baseball players playing in the majors every year. We can’t just extrapolate this number backwards, though, because a large number of those players are the same year after year. To get an estimate of the number of players who have ever played, we need to divide by the length of an average career. That’s another number that I’m sure could be looked up somewhere, but that would go against the spirit of this problem, so let’s pick a number– call it five years for a pro career on average. If anything, that might be on the high side, but it’s probably not wildly unreasonable– there are guys who play for 10-15 years, but there are also guys who get called up for part of one season, and that’s it.
Dividing the number playing now by the length of a career gives us an estimate of how many new players you have every year, which is 200. To turn this into a total number of players, we need a total number of years. They’ve been playing baseball for a long time, so let’s call it 100 years.
“Wait a minute,” you might well be saying, “they’ve got baseball records going back earlier than that.” Which is true, but we’re just looking for an estimate, here. And remember, our estimated number of players was calculated based on 30 teams in the league, which is a relatively recent phenomenon. They’ve added a bunch of teams over the years, so we’re significantly overestimating the number of players per year for the earlier years. So if we underestimate the number of years baseball has existed, that’s not going to be that big a deal.
So, at 200 new players per year, for 100 years, we’re looking at 20,000 players as our estimated count for the total number of players who have ever played (and thus might be said to have tried to hit 600 home runs). Which is pretty close to Valentine’s 17,000.
So, while that figure is probably a made-up number to sound good, it is, pardon the expression, in the right ballpark. 20,000 is probably too high, but not by that much. It’s almost certainly the right order of magnitude– there’s no way there have been 200,000 major-league baseball players over the years, and the number has to be higher than 2,000, because there have probably been 2,000 guys playing at least one game in the majors in just the last ten years.
And there’s your demonstration of Fermi-problem estimation for the week…