I just heard that Monty Hall died at the age of 96. He was a game show host famous for a simple probability problem that gets people riled up. His show, Let's Make a Deal, had three doors. Behind one door was something very desirable - like a new sports car. The other two doors had, well, something like a goat. You were asked to pick door number one, door number two or door number three. At that point your chances of worrying about the color of your new car were one in three. After you made your choice Monty would open one of the other doors revealing a goat and would ask if you wanted to switch.
People tend to think it's a coin flip at that point, but that's not thinking it through. Monte has perfect knowledge of the situation and has added some information to the puzzle with is reveal. The right strategy is to switch - always. Your chances will have improved to two out of three.
One way to think about it is if you picked, say, door number one, the chances of it wining are one in three and the chances of each of the other two are one third each. This means the chance of it being in either two or three is two thirds. Monte, by opening door three (for example), has increased the likelihood of it being behind door number two to two thirds. He added information.
always use extra ground truth when it is available!
You won't always win, but if you play a large number of times, you'll win about two thirds of the time. You can write a little program to convince yourself of imagine a game where Monte has ten thousand doors. One car, 9,999 goats. You choose door number one. Monte opens all of the doors except, say, 1,729 and asks if you want to switch. My guess is your intuition will do the right thing:-)
Comments