Monty Hall

Daniel — Wed, 04 Jul 2007 18:49:09 +0000

One of my favourite thinking puzzles is something called the Monty Hall Problem. The great thing about this particular problem is the extent to which people are willing to go to argue their point, and the sheer belief (and associated disbelief) they hold that their solution is correct (and that yours is wrong!). I love the argument and discussion that goes on whenever Monty Hall is mentioned.

The Problem

The basic idea is that you’re a contestant on a game show.

There are three doors in front of you
One of the doors has a car behind it
The other two both have goats behind them
Monty Hall, the host of the show, knows what is behind the doors
You have to pick a door
Once you have picked a door, Monty will open one of the other doors and reveal a goat
Monty will then give you the option of sticking with the door you have picked, or switching to the unopened door

The goal of the game is to end up with the door that has a car behind it.

The Solution

So, what would you do?

The first (and most obvious) answer is that it doesn’t matter whether you switch or not. The chance of you winning the car must be exactly the same.

50%

1/2

The actual answer is that it’s better to switch. You are going to win the car twice as many times if you switch doors than if you stick. That is,

The probability of winning if you stick: 1/3

The probability of winning if you switch: 2/3

Unintuitive

It took me a long time to come to terms and agree with this solution. I didn’t want to believe that it was better to switch. I remember thinking to myself,

If there are two doors left, one of them has a goat behind it, and the other has a car behind it, the probability of picking the door with the car has to be exactly the same as that of picking the door with the goat. It doesn’t matter if you switch or not.

But that is wrong. I had to draw a whole heap of diagrams to convince myself of this (drawing a decision tree eventually convinced me).

Thinking

What I like about this kind of problem is that not only does it take a lot of thinking to come to terms with the solution, but it also takes a lot of thinking if you want to convince the doubters. When I first came across this problem, we ended up with a completely wasted afternoon at work while we discussed and argued the possible solutions.

Being able to think about and explain such a counter-intuitive problem, and eventually convince people of the correct answer, is great fun and a really enjoyable brain work out.

Agree

So, do you agree with my solution above? Is it worth switching doors, or doesn’t it matter?

What would you do?