Last night, Stephen Colbert gave a speech mocking President Bush at the White House Correspondents Dinner, which Bush attended. The video (a 3 parter on YouTube on Google Video) is pretty long, but quite funny. It's getting some "blogosphere" buzz... will it go viral like Jon Stewart's "Crossfire" appearance?
NBC has a game show called Deal Or No Deal in which the contestant picks one of 26 suitcases containing amounts of money ranging from $.01 to $1,000,000. After picking the suitcase, the contestant slowly eliminates the other suitcases, revealing their amounts. The contestant is periodically offered a "deal" to trade in their suitcase for some amount of money.
If you've eliminated a lot of the high paying suitcases, you get offered a low deal. If you've eliminated all but the most valuable suitcases, you get offered a pretty sweet deal. If you decline all the deals to the very end, you get the amount of money in the mystery suitcase.
It seems like a clear case of probabilistic expectation, so I was curious if the "deal" offers were just the expected value of the contestant's suitcase; I don't watch the show often and I don't normally carry a calculator with me, but I was watching last night and had my laptop with me, so I decided to quickly calculate some odds.
When we tuned in, the current contestant was down to 3 suitcases: $.01; $400k; and $750k. Ignoring the one cent suitcase, that's 1/3 * 400k + 1/3 * 750k or an expected value of $383k. Her deal? $375k. Pretty close. (She took the deal; her suitcase was worth $400k, so she only lost out on $25k. Chump change =).
One contestant later, they were down to 8 suitcases with one worth $750k. The offer given to the contestant was $21k. Now $750k / 8 is almost $100k, so the deal value was clearly below expectation. The contestant eliminated two more (low value) suitcases, and her deal got bumped up to $63k. One more low value suitcase down and her deal went up to $96k (her suitcase now had expected value $150k). She took that deal.
They then walked through what would have happened had she had continued the game. She kept eliminating suitcases until there were 2 suitcases (the one she'd picked and one last mystery case) and they said her offer would have been $400k. $25k above expectation.
So it looks like the game has an increasing expectation multiplier that actually hits >1. I guess if I'm ever on the show, I'll work my way down to 2 suitcases and then take the deal -- proving once and for all that mathematicians make piss poor gameshow contestants. Now if only they'd bring back Let's Make a Deal...
