Sunday, January 09, 2011

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.

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At Tuesday, January 11, 2011 3:32:00 AM, Blogger Vic Ferrari said...

Wow, that's cool. Intuitively I was sure the answer would be .500. Sample size is the culprit, it took me a while to get my head around that.

How many rookies are you supposing, Phil? A league full of rookies? If you supposed just one rookie per team, or having just one team running with this scheme, that would have a bigger effect.

At Tuesday, January 11, 2011 10:15:00 AM, Blogger Phil Birnbaum said...

Hi, Vic,

It's not me supposing, it's Landsburg! He posts the solution for various numbers at his blog. You're right, the answer is farthest from .500 when it's just one rookie.

At Thursday, January 13, 2011 3:00:00 PM, Blogger Vic Ferrari said...

Douglas Zare's answer
uses clever math. That saves a lot of computation. In fact it seems that Zares post was Landsburg's inspiration.

I think Landsburg's puzzle is a bit misleading. A kingdom with 5 or 10 families is a bit of a nonsense, that's a hamlet at best, not a kingdom. And of course with bigger populations the answer ends up so near .500 that it doesn't sensibly matter.

Your baseball analogy fits the interesting range of Zare's math much better I think. Perhaps that's why it's created less controversy and debate :)

Another way of framing it would be:
Total rookie AB (or total children born) is a simple geometric distribution with p=.500 ...

So regardless of the size of the population, we expect to see and aggregate average <.500 exactly 50%of the time. Just as we would taking random samples for a geometric distribution. And >=.500 exactly 50% of the time.

Of course with smaller populations the chances of the average being exactly .500 is nontrivial (with one player it's 25%, with two players it's 18.375%, etc) ... that's what is tricking our intuitions, I suspect.


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