Chipper Jones' chance for .400
Over at Baseball Prospectus, Nate Silver gives us an estimate of the chance that Chipper Jones will wind up hitting .400 this year. (Subscription required, I think.)
Silver starts by estimating that, before the season started, our best estimate for Chipper's talent was a normal curve centered around .310. But, now that he's hitting .419 so far this year (or whatever it was when Silver wrote his analysis), the best estimate is now a normal curve centered at .345 or so.
Then, it's straightforward: figure the chance that Chipper is a .300 hitter, and multiply the chance that he'll hit well enough as a .300 hitter to finish him above .400. Repeat for .301, .302, and so on. Add up all those numbers and you have his probability.
Actually, Silver didn't quite do it that way: he did it by simulation, 1000 repetitions of picking a random talent, then playing out the season. I think a thousand reps isn't very many to get a precise estimate, but it should be unbiased, at least.
Silver comes up with a probability of 12-13%.
However, there are a couple of problems with the analysis, that Tom Tango nails over at his blog.
First, the estimate of Chipper's talent shouldn't be normal. It should be biased towards the left. That is, even if your best guess is that Chipper is .345, you should give him a much better chance to be .335 than .355. Silver makes them equal. [UPDATE: Tango's comment, and further reflection, have convinced me that this criticism is not correct, and that Silver's symmetrical distribution is in fact correct. See the comments.]
Second, bumping Chipper from .310 to .345 on the basis of a couple of hundred at-bats is too big a jump. His talent almost certainly should be centered lower than .345.
As Tango points out, when you're looking at low-probability events, a small change in assumptions makes a huge change in probabilities. And so Silver's probability estimate is probably too high – much too high.