Mathstina, in a post from August 25, put put a video from the Australian version of Deal Or No Deal which showed a spectacularly unlucky contestant, a contestant unlucky enough to inspire word problems. I quite like game shows, partly because I was a kid in an era — the late 70s and early 80s — when the American daytime game show was at a creative and commercial peak, when one could reasonably expect to see novel shows on two or three networks from 9 am until 1 or 2 pm, and partly because they give many wonderful, easy-to-understand mathematics problems. Here’s one I based on the show and used as an exam problem.
First, the basic rules of the game: there are a certain number of suitcases, each containing a set amount of money. The Contestant has one. The rest of the suitcases have their contents very slowly revealed over the course of the game. Periodically the Contestant is given the choice to accept the sure thing of the Banker’s offer. As the non-Contestant’s suitcases are opened and their contents revealed, the Banker’s offer rises or lowers based on whether more low-value or high-value suitcases remain unopened. Meanwhile people at home yell at the TV.
Suppose that at some point in the game, there are five unopened suitcases (the Contestant’s and four unselected ones). The amounts not yet revealed are $1, $10, $7,500, $25,000, and $35,000, so in the Contestant’s suitcase is one of the five. The Banker offers $11,750 for the Contestant to give up whatever the contents of her suitcase are and walk away.
Should the Contestant take the sure thing, or should she hold out for her suitcase’s contents? And, better, why?
nebusresearch 
Because the Professor formulating the problem has described the Contestant as ‘spectacularly unlucky’, the Contestant should take the Banker’s offer. My reasoning is thus:
We may reasonably assume the Professor to have full knowledge of the outcome of the game. For the Contestant to be descibed as ‘spectacularly unlucky’, we may reasonably assume that she got the worst possible result, and ended up with $1. Since that’s the lowest outcome possible, we may safely assume that under the rules of the game the Banker would never offer that amount. Therefore, the Contestant must have gotten the worst possible result by refusing all Banker’s offers, and hanging on to her suitcase.
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I’m sorry; I was unnecessarily confusing there. The video shows a contestant who managed to pick off the top four prizes, eliminating the biggest payout, in descending order, so the possible offers from the Banker (or the amount in the suitcase) could be assumed to have shrunk as rapidly as possible. The contestant was remarkably unlucky not in picking the lowest-value suitcase (that’s as likely as picking the highest-value suitcase, after all) but in picking the highest-possible-value suitcase for elimination four times in a row.
The question I put out about the five unrevealed suitcases was intended to be a separate problem, not the one faced by the video’s contestant.
I don’t know whether by the construction of the game the Banker or the Host know which suitcase holds the greatest or the lowest values. My suspicion is that they would not because the fewer people know the money amounts and the less this information is communicated, the harder it is to rig the show’s outcome.
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Well, for the purely mathematical, playing-the-odds answer: The total money up for grabs is $67511. Divided by 5, that means the average money left in each case is $13502.20. So the contestant should hold out, given that $13502.20 is larger than $11750.
–Chi
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You’re correctly calculating the expectation value, or in other terms, the average payoff for the Contestant who in this situation sticks to her suitcase rather than taking the sure thing.
Arguably, therefore, sticking with the suitcase is the correct thing to do.
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I’m with Chiaroscuro on this one.
Also, I want to be on Deal Or No Deal solely so that I can run through the numbers as quickly and purely-randomly as I can, because I can’t stand watching that show which is nothing but people talking about their supersitious reasons why they’d be taking one number over another. If they insist on patter then I’d come up with reasoning about how much I HATE each number and why none of them could possibly be a winning number. like, SCREW 13, that’s how old I was when my grandma who taught me chess died!
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Ah, now … I don’t like the post-Millionaire trend of giving contestants unlimited time to reason out their answer, but I also understand that for a show like Deal or No Deal, where there’s literally nothing to do other than for the Contestant to say why she or he is picking this, that the explanation is … well, no, the suspense is the meat of the show, but there’s not suspense unless there’s something the audience is waiting to see revealed, and if you rush things there’s no waiting.
I suspect that the show’s producers would be happy with you coming up with lunatic reasons for each number selection. (I’m reminded of a comic who said that if he were to be on Millionaire he’d want to come up with ludicrously wrongheaded chains of reasoning before picking the correct answer, eg, concluding Pearl Harbor was in 1941 because he remembers it was before Lincoln was shot but after the invention of Tamagotchis.) But rushing through just wouldn’t do; the audience has to get to know you through the explanations.
(I also don’t like the Millionaire-influenced design of game show sets as dark, technophobia-induced pits with heavy bass playing. The Australian Deal or No Deal set better fits the tone of being a lighter, warmer show. It should be more cozy than oppressive.)
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