Saturday, September 23

Deal? No! Probability

So I'm flipping through the channels right now and I've come across 'Deal or No Deal'. Basically, there are 26 briefcases and each one has a certain amount of money in it ranging from $0.01 to $3 million. The contestant picks one in the beginning and then eliminates the others. If he's lucky then he's eliminating the small values. Periodically the 'banker' calls and offers to give the contestant a certain amount of money to buy him out of the game. There's no knowledge involved, rather just guessing and probability.

That brings me up to the present. I'm watching the show for about the 3rd time and I think I know how to beat the system. Don't rely on the audience or Howie. If you're sitting in the audience at 'Who wants to be a millionaire?', would you type in the right answer when they poll the audience? Also, Howie is constantly telling the contestant how huge an offer the banker is giving him.

Right now, there's a rather annoying family from NY, I think. I can tell because they almost seem like caricatures like the characters from that annoying SNL skit with Jimmy Falon and Rachel Dratch.

From Stats 221, I know the simple math that I think would allow you to beat the system. It's called 'expected value'. Since each value occurs once and just once, the probability simplifies. In a nutshell, when the game starts, add all the values together, $3,418,416.01 (in a 1 million dollar game) and divide by the number of values, 26. Randomly playing, one should then expect to get $131,477.54. As you eliminate amounts, add the remaining and divide by the number of values and that's what you should expect.

As I'm watching, the banker just offered him $675 K. He had $1, $400K, $750K, $1M, and $3M left. He took the offer. The expected value was $1,030,000.20. The show made money, nearly $400K. By the way, they went through the remaining cases and he had, in fact, picked the $3 million case.

If I played, I'd probably be the first to whip out a calculator and do some math to decide. It'd be a matter of playing until the expected value started to decline (meaning I started to eliminate larger amounts faster than the smaller amounts), or the banker made an offer close to the expected value. But even then, nothing is for certain since when he accepted the offer, though he was expecting more than a million, it was as likely that he'd end up with $1 as $3 million.

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