Here's a post from the Jeopardy forum:
I took the liberty of e-mailing Dr. Levine to ask him about the reasoning behind his 25-million-to-one estimate. Here’s his reply (posted with his permission):
Quote:
They said that in the final stage each player had roughly 5000 dollars and could bet any dollar amount. Assume they all get the questions right. Given the bet of the first player, if the second player chose at random he'd have roughly a 1/5000 chance of matching the sum of the amount bet by the first player plus his initial cash; the third player then has roughly a 1/5000 chance of matching the first two, so 1/5000 * 1/5000 = 1/25,000,000.
From what they told me, the players were considerably more clever than just betting at random, so if we take that into account, they had a higher chance of tying. For example, they could guarantee a 3-way tie by each betting everything then intentionally getting the questions wrong. So the question being answered was: approximately what were the chances they could tie if they bet randomly, they started on a near equal footing, were good players and tried to answer the questions right.
Frankly as an expert Jeopardy! player you can probably give a better answer than I - but I guess they thought it would sound better coming from an academic.
As I pointed out in my return e-mail to Dr. Levine, it’s clear from his reply that he is not a regular viewer of the show and that his estimate was not based on any kind of sophisticated game-theoretic analysis; he simply gave an abstract (and rather straightforward) answer to a set of conditions posed to him by the show’s producers. In fact, those conditions are unrealistic in several respects: (1) player scores entering FJ are generally considerably higher than $5000; (2) barring “cute” DD wagers in the earlier rounds, scores are normally multiples of $100, reducing the number of combinatorial possibilities to be taken into account; (3) players’ pre-FJ scores are not independent, because the total available clue value is limited—an unusually high score by the leader normally implies low scores by the other two players, again cutting down on the number of possibilities to account for; and most important, (4) players’ FJ wagers are strategically determined and not purely random.
All this is not to find fault with Dr. Levine’s analysis but rather with the conditions he was given to base it on. It’s a sad commentary on the state of mathematical acumen in our society that the show’s producers feel they need to consult a highly credentialed professor of econometrics to tell them that 5000 squared equals 25,000,000—and then that answer gets repeated uncritically all over the media as gospel truth because, after all, “a mathematician” said so.
Monkey excrement indeed—but Dr. David Levine of Washington University is not the monkey, and the excrement is not his.
