17 January 2009

Deal or No Deal

Howie Mandel LIES. I was laying on the couch the other day and my wife has Deal or No Deal on TV. Howie says something like "There is a one-in-four chance (25%) that your case contains $1,000,000!" Exciting, but wrong. Let's start at the beginning. Deal or No Deal is a game show where the contestant first picks one from among many briefcases. Most of the cases contain a small amount of money, a few contain a lot. The contestant knows the "list" of amounts that are inside the cases, but not which case contains what amount. After picking the first case, the contestant gets to open a series of other cases and learn what is inside. With each opened case the contestant gets an offer; they can trade their case for what is offered, or after they open all the other cases first, keep the case they first selected. Now where was I ... Oh yes ... a contestant is down to 4 cases, including their original pick. One of the remaining cases contains a million dollars and then (here we go) Howie says something like "There is a one-in-four chance (25%) that the contestant's case contains$1,000,000", and Howie is wrong.

Let's pretend I am playing and start with 10 briefcases. One contains $1,000,000 and the nine others contain$1 each (we could vary this like the game, but I am simplifying). Not knowing any better, I pick a case randomly, and I have an expectation of a 1-in-10 chance it contains $1,000,000. Next, Howie lets me open 6 other cases (skipping the offers). If I am lucky and none of those 6 contain the$1,000,000 amount, then 4 cases remain and one contains the big money. If I could pick one randomly NOW, there is a 1-in-4 chance it would be the $1,000,000 case, BUT that does not change the odds for my original pick. My first pick is still a 1-in-10 chance, but the other 3 are now 1-in-4 chance(s). The difference is now I know more than I did at the start; I now know 6 cases that do not contain the big money, and this information changes the value of the remaining cases. If this seems confusing, then you are in good company, because this problem has stumped a lot of very smart people. This is a famous problem of conditional probability (famous in Math/Stat circles anyway) known as the Monty Hall Problem. I changed it around a bit to frame it as "chance of$1,000,000" rather than the usual expected value, but it is the same problem.

See The Monty Hall page for a nice online demonstration. One more thing: Howie Mandel is no Monty Hall (sorry Howie).