wrzesień 2009 - The door puzzle

The door puzzle

You may have heard this before. After all, it is a well known puzzle. If you haven't, you may be in for a surprise.

The task: a game-show host asks you to choose one of three locked doors. The prize (a car, stack of banknotes or whatever takes your fancy) is behind one of them. You have no clue which door is the right one. Now that you have chosen, the game-show host opens one of the remaining two doors to show there is nothing on the other side (sometimes there is a 'booby prize', like a wooden spoon). At this point the host asks you to make another choice. Either you can stay with your original door, or swap for the other, unopened, door. Which do you choose?

To make things clearer, the host knows which door has the prize behind it. When he opened one, he knew there was nothing behind it. This is a key point when you come to decide the best strategy.

Now here's the important question. Do you have a greater chance of winning the prize if you swap doors, a smaller chance, or does it make no difference whether you swap or not?

What does your instinct tell you? What about the mathematics of the choice? Will you change to the other closed door and try your luck there or will you stay with your first choice? Will you toss a coin to decide, particularly if there is no difference whether you change or not?

A guaranteed win is impossible. This is a game of luck, but is it all luck? Snakes and ladders (mentioned in the exercise for the previous entry) is a game of pure luck, while chess is pure strategy. Two of my favourites are backgammon (watch this space for more on that) and poker, which are a mixture of the two. So what about the door puzzle? Is it as random as a primary-school board game, as logical as the sport of kings, or a luck-skill combination?

Try it with two people and three playing cards first. One of you has to be the host, so they have to know which card is the prize. Try each strategy several times and see how often you win. When you get to about your twentieth go with each of the two methods, you should spot a pattern. Did it surprise you?

If it did, you are in good company. The puzzle has even confused statisticians, scientists, mathematicians and accountants. The reason for this is that your instinct may tell you one thing, but even when you see the maths behind the answer, instinct remains strong. Here is one way to get your head around the solution if it's still confusing: one of the host's questions is irrelevant.

Which one and why?

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