[quote name=\'Matt Ottinger\' date=\'Dec 21 2005, 12:57 AM\']Oh, if only we had a professional math teacher with an understanding of game show theories and strategies on this board. Even one who doesn't post much.
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I thought I heard my phone ringing, but decided it was probably that pesky banker... :-)
The basic question has already been answered; if there are four cases left (three on stage, one for the contestant), then all probabilities should be based out of four suitcases. Thus, if there are 3 out of 4 that will improve the contestant's situation, the odds are with him or her to continue on in the game.
But that's where this game gets tricky. First, there is the concept of the contestant's mathematical expectation in the game. The expectation is computed by multiplying all possible probabilities and corresponding payouts and then summing them. For example, if someone has four cases left with amounts of $75, $500, $50,000, $500,000, then the contestant's expectation is (1/4)(75)+(1/4)(500)+(1/4)(50000)+(1/4)(500000) = 137,643.75. Since the bank offer basically equaled that expectation (rounded to the nearest thousand), an argument can be made for the contestant taking it.
Then again, that's not totally correct, because the truer expectation should be based not on what's in the player's suitcase, but rather how much higher the deal could potentially go. That's difficult to predict, of course, since the bank does not always offer something close to the exact average of the remaining amounts... but let's assume it does. Then, if the contestant chooses one more suitcase, there's a 1/4 chance of uncovering each of the amounts; the resulting deals would end up approximately as follows:
If $75 is chosen, the deal will be about 550500/3 = $183,000
If $500 is chosen, the deal will be about 550075/3 = $183,000
If $50000 is chosen, the deal will be about 500575/3 = $167,000
If $500000 is chosen, the deal will be about 50575/3 = $17,000
Thus, the expectation if the contestant chooses one more case is:
(1/2)(183000)+(1/4)(167000)+(1/4)(17000) = $137,500. In fact, if you work this scenario out in general, the expectation will always be the same as it is if you just consider the individual cases as I did originally above. However, this assumes that the bank will offer something close to the average. Still, I (and some others of us) have noticed that the bank offers do tend to approach the true mean as more and more suitcases are removed, thus confirming that the producers want the contestant to stay in the game for as long as possible (since the deals do tend to get better from a percentage sense).
But perhaps the most important consideration of all has nothing to do with complicated mathematics; it has to do with the raw emotion of it all... namely, if you're standing there with essentially $138,000 to call your own, how hard does that become to turn down? In that case, other factors work their way into the expectation; it may not just be a case of how much money you stand to lose, but what sort of an effect that might have on you. Of course, the same can be said about what sort of an effect winning more money would have as well, but the point is that suddenly any sort of expectation calculation takes on a whole new perspective; it's not just about the numbers any more.
