Bad Luck on Deal Or No Deal

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?