My wife caught me watching Deal or No Deal last night after I'd vowed never to watch it again. She threatened to go public, but I told her I was watching it with a spreadsheet open, hence making it educational. So, I confess to watching.
There are several bloggers out there attempting to determine what the formula is the banker uses to make his offers. In searching through them, I learned that Howie Mandell is a germaphobe, and that's why he doesn't shake hands with contestants. He does "the rock." He hates touching people. He also gets mad when dumb contestants miss out on much-needed cash by pressing their luck.
So, a fun game you can play at home is "touch Howie," count how many times he physically touches someone and does so without cringing. Take bets on whether or not he runs from contestants when they win big money and want to hug him.
So, here's how to play along at home to determine whether or not the player should take the deal: You add up the probability of being left with each individual case. At the start of the game this is ((1/26)*.01)+ ((1/26)*1)+((1/26)*5), etc.
This ends up being the average of all the values on the board (much quicker to figure). That is the contestant's expected value (payoff) of continuing to play.
This guy has a nice, clean Excel sheet that lets you do this quickly.
If the banker makes an offer higher than the expected value, then the contestant should take the deal. So, if you're ever on the show you should always average the values of whatever is left on the board and wait for the banker to make you a higher offer.
Interestingly, the formula for the banker's offer appears to have changed recently. The banker previously would make offers lower than the expected value until about the 6th round, when the offers would go above the expected value, enticing the player to quit playing. However, it now appears that the banker always offers a number lower than the expected value, and that the offer might be a random number within a certain percentage. Perhaps this is because they want someone to finally win the $1 million? Perhaps it is because they want to extend the game and minimize contestants and therefore minimize their total payouts?
On WikiAnswers, someone has posted a formula that they say will get you within the banker's offer by $10,000.
Last night's game featured another unusual twist. The contestants had to wager their winnings on the Deal Wheel, a game where they had a 35% chance of halving their winnings and a 65% chance of at least doubling them. I think the contestants should be able to opt out of gambling their winnings as such. Throughout the show, Howie made it sound like contestants would triple their winnings automatically. Instead, there is only a 21% chance of tripling them (if we counted correctly).
So, I'll now be testing the various formulas people have posted online for accuracy.
Note: You can also read the entertaining blog of The Banker. He berates various commenters and causes an angry furor.
I'll never watch it for fun, only for mathematical intrigue. (Except wagering on "Touch Howie" might be kind of fun).