# Baseball and the Achimedean property

Are there any batting averages which are impossible for a player to achieve over the course of a single season?

I initially thought that .001 would not be achievable. But most of the records for at-bats in a single season are above 670, and 1 hit in 670 at-bats = .00149, so it would get rounded down. But it's close! If nobody had more than 666 at-bats (81st on the record list), or if we only rounded up, then .001 would be impossible.

But as a practical matter, nobody who went 669 at-bats without a hit would remain in the major leagues. Perhaps nobody hitting close to .999 would be permitted anything but a walk, so 666/667 does not seem practical either. And the actual distribution is obviously nonuniform, so real batting averages don't contain anywhere near the whole ~10 bits of information in three digits.

So perhaps I should be asking:

* What three-digit season batting averages have never been achieved? If we exclude those without enough qualifying at-bats, then it looks like at the high end we have gaps:

1.000 - .486
.484 - .458
.455 - .441
.439 - .430
.428
.425
.423 - .421
.419 - .416
.414 - .413
.411
.405
.400

(but many of these use the 1887 accounting for walks.)

Finding the low end would take more patience than I care to expend.

* How many bits of entropy *are* present in a three-digit season batting average drawn from across baseball history?

(Icon irrelevance: two of my four poker-player icons have been charged with defrauding poker players. Guess I won't be using those for many future posts. What does this say about my choices?)
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