There are three doors (1, 2 and 3) with one prize hidden behind each one. There is one car and two goats. Obviously, the idea is to get lucky and pick the car. Equally obviously, there is a one third chance of getting the car.
You can pick whichever door you like, for example door 1.
Monty knows where the car is. He has a think. He opens door 3 and reveals a goat.
He then offers you the chance to switch from door 1 to door 2 (or to stick with door 1).
Do you switch from 1 to 2?
When I first faced this problem I said no. I would stick with door 1. I reasoned that now there is a 50% chance of getting it right, and I might as well stick with what I put down first. It turns out that I was wrong.
If you switch to door 2 there is a higher probability of getting the car!
Wikipedia is full of differing explanations for the maths behind it, but I'll explain it in the way I reasoned it out.
At the start of the problem there is a one third chance of picking the car and you pick door 1. Thus there is a two thirds chance that one of doors 2 and 3 have the car. Therefore there is a two thirds chance that when Monty chose the door to open, he had no choice (he could not open the other door as that would have revealed the car).
There is only a one third chance that neither 2 or 3 had the car. In this case Monty could have picked either 2 or 3 to open.
Thus because there is a 2/3 probability that Monty had to open 3 and there is only a 1/3 chance he could have chosen either, we should switch to the one he did not open: door 2.
It took me a long time to work that out. Please let me know whether my explanation is adequate...
Recommended listening:
Fast Car by Tracy Chapman
Hi (this is Silky). The phenomenon actually does contain some behavioural economics. When experiments were run on this they found that most people stuck to their original choice; the reason so few people are willing to switch to the other door is because of an endowment effect; you become attached to your choice of door!
ReplyDeleteA simpler way to explain it is this:
ReplyDeleteIf you originally choose the door with the car (1/3 probability), then changing door will always make you lose.
If you originally choose a door without the car (2/3 probability) then changing door will always make you win.
In other words, if you always change door, you will win 2/3 of the time.
Thanks Silky!
ReplyDeleteVery helpful