Here’s a puzzler for you.
Let’s say I have two tiles: one painted blue on both sides, and the other painted blue on one side and white on the other. I mix them up out of your view, then randomly place one on the table. You look over and see a blue-sided tile face up. What are the odds the other side is also blue?
The natural inclination is to say it’s a fifty-fifty proposition. But in reality, there’s a two-in-three chance you’ve got blue on the underside.
If I were to ask “What are the odds you have the tile with white on one side?”, then it would be even odds. But that’s not the question I asked. Imagine now that the three blue tiles were marked with an A, B, and C. A and B are back-to-back, and C is on the flip side of the white tile.
We know there is a blue tile facing us on the surface, but it could be any one of the three blue sides.
If it is “A”, then the hidden side is a blue (B)
If it is “B”, then the hidden side is a blue (A)
If it is “C”, then the hidden side is white.
That’s a two-out-of-three chance of a blue tile underneath, flying in the face of the average intuition. If you didn’t catch that right away, don’t worry. It’s a tricky proposition, because there are elements you can’t see, and you’re not really sure what you are seeing.
It’s the unseen that makes “Deal or No Deal” so compelling at first. There are no patterns, no questions, no Whammies. Every person gets the same game. You make a random draw, and as you eliminate the blind boxes you didn’t pick, you learn more about what is left. Along the way, you are offered a buyout to walk away.
I’m not going to bother trying to break down the formula for you, but I find it interesting that as a purely mathematical exercise, it would be a colossal failure. For such a system to work, you would have to have players acting as purely rational economic beings. And as we all know, game show contestants do not meet the criteria of “purely rational.” In fact, in “Deal or No Deal,” the producers stack the deck with both a studio audience and family members who exert peer pressure to open “just one more briefcase.”
Such a system is way to messy to figure out… unless you have a test lab to play with. But how could you get tens of thousands of average people to play through the game for no money, just to see where the average breaking point might be? If you knew that, you could set up a system for the Banker on that show, and minimize the Banker’s payout. I suspect the tens of thousands of players of the online version are the guinea pigs.
(If you’re interested, I posted an analysis on the Freakonomics blog.)
Yes, the math gets really messy and unpredictable when human emotion and irrationality are introduced into the system. 
That does not mean that everything stays nice and neat when humans aren’t involved. Some systems, even those with a very simple number of elements, become quite chaotic in no time at all. Which takes us from Mandel to Mandelbrot…
[tags]Ike Pigott, Occam’s RazR, Deal or No Deal, Probability, Pop culture, Freakonomics, Mathematics[/tags]
