Landsburg's brain teaser, translated into baseball
Since Steven E. Landsburg posted his brain teaser a couple of weeks ago, many commenters have argued that the problem was not well described. So I thought I'd try rephrasing it in a baseball context, which might make it easier to understand. I'll try to include all the picky details. Here we go:
In a certain baseball league, every rookie is in the starting lineup on opening day.
Every rookie is an expected .500 hitter in all circumstances, and has an independent 50-50 chance of getting a hit in any AB.
Every rookie plays every inning of every game until he has a hitless at-bat, at which point he is immediately replaced by a veteran and never plays again.
What is the expected value of the overall composite rookie batting average at the end of the season?
[Details: 1. "overall rookie batting average" is total hits divided by total AB (so if there are two rookies in the league, a 1-for-2 and a 3-for-4, the composite batting average is .667 (4-for-6)).
2. "Expected value" is in the normal statistical sense. One way to explain "expected value": suppose someone offers a bet, where he promises to pay you a dollar multiplied by the composite batting average (so that if, like in the example above, the composite batting average winds up at .666666... , you get 0.66666... dollars). What is a fair price to pay for the bet, in dollars, so that neither party is expected to make or lose money?
3. You may assume, if you like, that no rookie will go the entire season without making an out.]
If I've done it right, this question should be completely isomorphic to the intent of the original. The answer and discussions at Landsburg's blog can be found here. My take was here.