A number of folks who know math better than I do sent me breakdowns on the odds involved in playing Deal or No Deal. They were all figuring slightly different scenarios and I don't understand any of them. However, this piece over on Wikipedia seems to know what it's talking about with regard to the odds. Here are some comments from e-mails I received. This first one is from David Oakes…
…for those of you who like to play along at home, calculating the likely value of the pot at any given time is pretty easy. Since the prizes are distributed randomly, simply add up the total money still in play, and divide by the remaining number of boxes. So at the very beginning, the game is worth $3,418,416.01/26. (If every contestant just took one box and stuck with it, on average they would go home with $131,477.54. Nothing to sneeze at, but not the "life changing money" of high ratings.) Coming down to 5 boxes containing 1,000,000, 0.01, 300, 1,000, and 750,000, the "expected value" of the prize is a little over $350,360. It would be interesting to compare these averages with the deals offered, to see how much the producers are willing to pay for ratings…
And this one is from Bob Heer…
Interestingly, if a player goes straight to the final step each time, one of the two remaining cases will be the million 1 in every 13 games, and probably getting an offer of over 500K, and better than 1 in 5 of having at least one 500K+ case (offer probably at least 250K, which I think is a lot more than anyone got in the first week). Not that jumping straight to the end blindly is a good strategy, since that also gives good odds of two small choices in the end (odds make for a good aggregate strategy, if you get to play an effectively infinite number of times, not if you only have one shot).
And Tom Galloway figures there's a 1 in 2600 chance of having three cases left and all being the top three money amounts. I think the key here though is that each contestant only has the one chance to play the game. The odds figured over multiple games don't become as controlling of what any individual might do when it comes down to those final decisions. I would think, above and beyond the cash itself, most would feel a powerful desire to not go home with the embarrassment of having turned down $250,000 to go for some higher amount, only to leave with five dollars. Even if the gamble seemed statistically logical, you'd still be the person who had a free quarter-million dollars in hand and threw it all away on national television because he/she wanted more. In other words, the tiny and large cash awards probably become significant in more than a strictly mathematical consideration.
One other thought occurred to me about the program…and it has to do with how the producers select contestants. That's key to most game shows — that they pick people who will be enthusiastic about winning but not to the extent of seeming out of control or hysterical. As a producer, you want players of whom America will say, "Oh, I want to see that person win." On this one, you also want people who really believe in luck, which I don't. If I went on Deal or No Deal, I'd probably pick Case #1 as my case and then as I had to open other cases, I'd just go in numerical order: 2, 3, 4, 5, etc. There's just as much chance of winning if you do that as if you start with your daughter's birthday and then pick her age and then pick the day of the month you met your significant other and so on. But picking my way would make it feel like the outcome was pre-determined (which to some extent it is) and that we're just going through the motion of opening cases and waiting for the banker offer to seem acceptable. That's pretty much all the game is but because the player seems to have a series of tough decisions to make along the way in choosing cases to open, it feels more like he or she is controlling what numbers are revealed. So we sit at home and think, "Oh, no! Don't pick 23. That's a bad number! Pick 6 instead!" Like that matters.
It really doesn't…but it feels like it does. Which is why it's all the more amazing to me that I got hooked on this silly little show and look forward to its return.