Deal or No Deal?

When I was looking something up on IMDB today, I noticed an advertisement for the gameshow Deal or No Deal?. The host is Howie Mandel, who has done some funny stuff before---I really enjoyed his show Good Grief, which lasted all of half a season. (He was also amusing in his occasional appearances on Politically Incorrect.) This show was well-known for being in extremely poor taste, which is one of the reasons I enjoyed it. In it, he works at Sincerity Mortuary, at which all sorts of antics occur. In fact, the original commercial for the show pretended it was a commercial for a reasl mortuary---We see an unctiously cheerful Mandel saying "At Sincerity Mortuary, we cry because we care! [pause] And we bury them because they're dead! [sobs into his hand]." All they were missing during their funerals and burials was the playing of "Another One Bites the Dust" (though there is a commercial for sound-proofing windows in which the guy driving a Heasre is blasting that song---I love that commercial!), so I'll make up for that by putting it in my will that that song is to be played at my funeral. (Yes, I really will be doing that. It's going to be the most tasteless funeral ever!)

Anyway, back to the point at hand.

So, the show is basically like the 'what's behind door # N' minigame of Let's Make a Deal except on crack. The actual show isn't all that fun to watch (because they belabor everything, just like Who Wants to be a Millionaire?), but there's a nice probability problem here (and some nice PS12ing to do). And I wanted to give it a chance and watch it once. Anyway, there are 26 models holding 26 cases, each of which holds some amount of money---ranging from 1 cent to $1 million. I believe that the median values (among the 26 "increments", as Howie Mandel called an individual value one time; oops) are $500 and $1000. The values below $500 don't really matter as far as the expectation is concerned (they can mid-game, but if you're ever in that case, you're pretty screwed anyway). There's also $5000, $10000, $25000, $50000, $75000, $100000, and then 200, 400, 500, and 750 K, and then $1 million. (Anyway, I think I got the numbers mostly correct, but this is at least enough info to see how many of the larger values are around.)

A contestant starts by picking 1 case. (I would so bring my D & D dice with me if I were a contestant on this show.) He/she must then pick 6 of the other cases to reveal. Those values are crossed off. Given the ones left (one of which is in the case the contestant chose), the contestant is offer a deal with an appropriately chosen amount. He/she can either take the money or not. If not, five more cases are revealed. The contestant gets an offer based on the amounts remaining, and can either take it or not. If not, four more cases are revealed. Then it goes to 3, and then 2, and then sequences of 1's. In terms of PS12ing, the key is how many big values are left on the board versus how many cases have to be revealed the next time. (It was amusing how they would root for the lowest value left to be in the chosen case even when the lowest N values would all have essentially isomorphic effects on the expectation.) The public at large has very fundamental issues inderstanding probability... (Daniel Kahneman won a Nobel Prize in economics for his research on that subject, by the way.)

In sum, this show makes a nice probability problem.