As a follow up to my simulation based approximate solution to the Gambling Machine Puzzle, here is the exact solution from mathematician Michael Lugo with a nice explaination.
An entrepreneur has devised a gambling machine that chooses two independent random variables x and y that are uniformly and independently distributed between 0 and 100. He plans to tell any customer the value of x and to ask him whether y > x or x > y.
If the customer guesses correctly, he is given y dollars. If x = y, he’s given y/2 dollars. And if he’s wrong about which is larger, he’s given nothing.
The entrepreneur plans to charge his customers $40 for the privilege of playing the game. Would you play?
Clearly the strategy is to guess that y > x if x is small, and to guess that y < x if x is large. Say you’re told x = 60. If you guess x is the larger variable, then conditional on your guess being correct (which…
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