Wednesday, May 31, 2006

How to Deal.

I confess to being fascinated by Deal or No Deal. It's a lot of fun to watch contestants get up there and, against their better judgement, bow to the pressure of a crowd shouting, "No deal! No deal!"

I also find it interesting that although the odds are quite good that a contestant will pick the $1 million case at the beginning -- 1 in 26, about twice the chance of winning $10 in the 6/49 -- the chances that the producers are ever going to pay out a full $1 million to a contestant are pretty slim. It may well never happen... I'm not sure if people watching at home are aware of this. And I wonder: if no one walks away with $1 million after a couple of seasons, will they stop watching? Will they tire of the tease?

Just how slim are the odds of walking away with the million?

Well, the anonymous probability experts at Wikipedia estimate that 1 in 130 contestants will walk away with the top prize. This is based on the rather spurious assumption that a contestant will keep opening briefcases as long as the $1 million briefcase and one of the briefcases with more than $200,000 are still unopened -- and then will take the 50/50 chance that she is holding the $1 million at the end, rather than the Banker's offer.

Personally, in theory, I would only say "No deal" all the way if I managed to leave the $750,000 and $1 million unopened all the way. (In reality, I'd probably cave long before that...) Assuming everyone thought like me, only 1 in 650 contestants would leave with the big prize. Even if the show ran five times a week all year-round (a possibility if NBC sinks any lower in the ratings), with an average of 1.5 contestants playing each show, there would only be a big winner every year and 8 months.

U of T probability professor and author of Struck by Lightning: The Curious World of Probabilities Jeffrey Rosenthal, who I interviewed for an article on this subject in today's Post (free link), estimates the odds of winning the big prize are much lower: 1 in 13,000. Here's his rationale, which I think got cut from today's article:
Well, obviously the banker will never *offer* $1 million, so the only way the contestant can win $1 million is if they refuse all deals and happen to end up with the $1 million prize. Now, *if* they refuse all deals, then their chance of winning $1 million is 1/26. But most contestants accept a deal at some point, and indeed the banker will tend to make more generous deals as the number of unopened briefcases diminishes if the $1 million prize is still available. Still, I would imagine that at some point in the future a player will be "brave" or "bold" enough to refuse all deals, and will end up with the $1 million prize. If I had to put a figure on it, I would say that perhaps one contestant in
five hundred will refuse all deals, so the probability of winning the $1 million prize would then become (1/500)/26 = 1 / 13,000, i.e. something like one contestant in 13,000 might win the $1 million prize. So it will take a long time, but if the show kept running forever (a big "if") then it would eventually happen.
I think the best idea is to ignore the nearly impossible $1 million and go on the show aiming to win over $131,477.54. That's the average amount you would win if you played over and over -- so as soon as you get over that, you've beat the average, which is beating the Banker in my book.

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