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?
Please feel free to comment below, especially if you disagree with me.
July 8th, 2007 15:07
I remember why the solution is correct and it only took me five minutes – ha!
Pick a better brain teaser
However, maths aside, I wouldn’t switch because conviction is easier to defend than doubt. (You know, if I was wrong and ended up with a goat)
July 10th, 2007 09:26
Mmm, clever.
The simplest way I can think of explaining it is:
You have 2 options, switch or don’t switch.
Switch- you win when either of the two other doors hide the car. Don’t Switch- you win only if your original door has the car.
The only other thing to add is- a surefire way to win is to pick the door Monty reveals has the goat. Then you will 100% get what you expect.
July 20th, 2007 09:48
I figured this out in 2 mins but to be honest are more worried that you wasted an afternoon at work debating this!
July 20th, 2007 09:59
Ok, an afternoon might have been a slight exaggeration.
We definitely had a lively debate though!
August 8th, 2007 05:45
It’s worth considering the same scenario as above, except that Monty state of knowledge is different.
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 doesn’t know what is behind the doors You pick a door Monty randomly opens one of the other doors and reveals a goat Monty give you the option of sticking with the door you have picked, or switching to the unopened door
Same solution?
August 8th, 2007 05:49
It’s worth considering the same scenario as above, except that Monty’s state of knowledge is different.
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 doesn’t know what is behind the doors.
You pick a door.
Monty randomly opens one of the other doors and reveals a goat.
Monty give you the option of sticking with the door you have picked, or switching to the unopened door.
Same solution?
(Moderator, please use this version that has better spacing and punctuation!)