Numbers game (contd.)

The Monty Hall problem requires clear, unbiased thinking as new information comes in

Mark Ronan

Getting the goat: The Monty Hall problem exposes a human tendency to stick with difficult decisions (Fir002 CC BY-SA 3.0)

My December/January Cosmos article (“Acting the Goat with the Greeks”) mentioned a puzzle (the Monty Hall problem) requiring clear, unbiased thinking as new information comes in. The original problem was this: you are given a choice of three doors, one of which hides a valuable prize. After you make your choice the host opens one of the other two doors, showing it was not hiding the prize. There are now only two closed doors and he offers you the opportunity to change your mind and pick the other door. Should you stick, or change?

A natural human tendency to stick with difficult decisions inclines many people to stay with their choice of door, arguing that anyone offered what is now a choice of two doors has a 50:50 chance, so there is nothing to be gained by changing your mind. Indeed, the sudden new challenge creates suspicion, confirming your original decision. But in fact you should change, because as I wrote in my original article, in 10 cases out of 30 you picked the right door. In the other 20 cases you picked a wrong door, and the game show host has now opened the other wrong door, so if you change you win. Another way of putting this is to say that your original one-in-three chance has not changed because the host’s choice of which door to open is dependent on your original choice — he never opens the prize door. The chances must add to 1, so the chance that the other remaining door is correct is now two in three.

To drive the point home here is an analogous situation without choices. Three prisoners languish on death row, and the supreme leader commands that one of the three will be reprieved. The warden knows who that is, but cannot reveal the name. So one prisoner (call him A), unable to learn his own fate, comes up with an alternative question. He asks the warden to reveal to him the name of one of the other two prisoners who will be executed and is told the name of B. Somehow he manages to get this information to prisoner C, thinking that their chances of avoiding death have both increased from 1/3 to 1/2. Prisoner C, however, smarter than A, knows that the warden’s response to the question gives zero information on A’s fate because it is a given fact that at least one of B or C must be executed. A’s chance remains at 1/3, but this new information shows that his own chance has doubled from 1/3 to 2/3.

If you still have doubts, return to the three-door question, but with 100 doors. Your chance of getting the right door is 1 in 100. In the other 99 cases you picked a wrong door, and as the host opens 98 of the remaining doors, he avoids the correct door, leaving that as the remaining option. Go for it! Of course anyone entering the game at this point will see a 50:50 chance, but you have additional information, and should use it.

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