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…

Comments

  1. #1 Tom
    August 18, 2011

    I thought from seeing “Fermi Baseball” I was going to see some spin-1/2 baseball statistics and an explanation of why the single state is allowed but the triple state is forbidden (it only becomes achievable because you couple through second base. Can’t go from home to third)

    Of course, there is no way for quantum mechanics to explain the infield fly rule.

  2. #2 Eric Lund
    August 18, 2011

    each team has something like 30 guys on the roster at any given time

    About 10 of those players are pitchers. The American League, which has about half of the teams in the majors, has the designated hitter rule, which has the practical effect that pitchers in the AL (almost) never have an at-bat. (The National League does not use the DH rule.) That correction alone slices about 15% off your estimate, giving Valentine’s 17000 figure.

    The DH rule has only been in effect for about 40 years, and some pitchers do move between the AL and the NL, so I might be overcorrecting. Then again, relief pitchers in the NL almost never get at-bats either (managers will usually send in a pinch hitter if a reliever is in the game long enough to get an at bat).

  3. #3 Ray Radlein
    August 18, 2011

    Back in the 1990s, I had the “Big Mac” MacMillan Baseball Encyclopedia on CD, giving me access to the complete records of every player who had ever played in any of the major leagues (National, American, Federal, and American Association). I could have popped it in, and done database search to count the number of records in the database, and given you an exact number as of whichever year.

    Alas, I’m not even entirely sure where that CD is, or if there’s even the slightest chance that I could get it to run under XP or Win7 or even Linux via emulation.

  4. #4 --bill
    August 18, 2011

    If you go to Baseball-Reference, you’ll see (on the front page) that there have been 17,651 major league baseball players.

  5. #5 Jamie
    August 18, 2011

    bill@4: but then Chad wouldn’t have got to have fun with math…

  6. #6 asad
    August 18, 2011

    I think a follow-up question is this: Is it reasonable to assume that every single player who has played in the history of baseball is actually attempting to hit as many home runs as possible? What fraction of them realize that they’re not power hitters and would be better off hitting strategically rather than trying to pound the ball?

  7. #7 Blaise Pascal
    August 18, 2011

    Interestingly, when I saw the title, I interpreted “hitters” differently, and started doing a mental Fermi calculation for total number of at-bats, not players.

    My estimate: There are 30 teams, and each team plays about 150 games a year (I know over 160 games are played per season now, but fewer were played earlier, as well as fewer teams earlier). That’s 2,250 games/year, for 100 years, or 225,000 games. A typical game is 9 innings, with a minimum of 6 at-bats per inning. Let’s call it 9 at-bats per inning, for 81 at-bats per game, time 225,000 games, or about 20 million at-bats.

    Does that seem reasonable?

  8. #8 nice_marmot
    August 18, 2011

    Respectfully, I would submit that both numbers (i.e., Mike & Mike’s and the Fermi estimate) are orders (plural) of magnitude too low.
    Millions of American little-league players have swung for the fences every summer trying to reach that milestone. I know I did. (My career HR stats: Zero.)
    That all but an infinitesimally-small number of us never made it to the majors (or even anywhere close) doesn’t make our attempts any less serious.
    Perhaps that’s part of the magical appeal of baseball: It helps us all stay in touch with our 8-year-old selves…
    Congrats to Jim Thome, by the way. In a game played by way too many McGwires, Cansecos, and A-Rods, he is a workmanlike player and an all-around good guy (http://en.wikipedia.org/wiki/Jim_Thome#Baseball_and_personal_life).

  9. #9 Dave
    August 20, 2011

    I would appreciate a discussion of what game circumstances dictate a suicide (runner leaves 3rd base at the pitch) rather than a safety (runner leaves base at the batter’s bunt) squeeze. Surely there is a physics basis to this vexing issue.

  10. #10 Dave
    August 20, 2011

    I would appreciate a discussion of what game circumstances dictate a suicide (runner leaves 3rd base at the pitch) rather than a safety (runner leaves base at the batter’s bunt) squeeze. Surely there is a physics basis to this vexing issue.

  11. #11 Rob Ryan
    January 3, 2012

    I respectfully express my congratulations (OT I know) to Jim Thome who played for my beloved White Sox for a few years. He seems to be nothing but a great guy in every way. I know that, as late as last season, people questioned Thome’s Hall of Fame credentials. If he’s not inducted in his first vote, the Hall of Fame becomes a farce to me.

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