American magazine columnist Marilyn vos Savant once posed the following question, submitted by a reader:
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?
A large majority number of readers answered that question with “No”. Many assumed that the host’s action in opening one of the three doors changes the probability of the remaining choices, so that there was now a 1/2 chance that the door you selected is a winner, meaning you should probably stick with your original decision.
The answer, surprisingly though, is yes, it is to your advantage to change your mind:
Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance…
The winning odds of 1/3 on the first choice can’t go up to 1/2 just because the host opens a losing door. To illustrate this, let’s say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you’ve chosen, we’ve learned nothing to allow us to revise the odds on the shell under your finger.
This answer provoked enraged responses, many from mathematicians who were certain vos Savant’s answer was wrong. Even in the face of proof that she was correct, people insisted that it could not be so.
It’s fascinating to observe cognitive error in action as people fiercely refuse to change their minds, despite evidence to the contrary.
For a discussion of why this is so, read “Are You Smart Enough to Change Your Mind?” from 7P Productions, which links to the vos Savant article and to a fun interactive version of the Monty Hall problem, goats included, which includes an explanation of how the game works.
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