The other week we went to the Peak District and spent the evening as family in the pub. It's nice to spend time together, I was telling a friend, we seldom find the time now the older boys are teenagers. Some evenings, they barely say a word.
'What did you talk about,' he asked. 'Football? The X factor? Girlfriends?'
'Err, no. We discussed the Monty Hall Problem.'
"I won't ask,' he said.
We're a strange lot, but my kids are used to it by now. Michael was especially pleased as he's studying The Curious Incident of the Dog in the Night-Time, which happens to mention the problem - and evidently nobody in his class could understand it. And I was chuffed because for the first time I managed to explain it clearly enough so that somebody did.
Let's see if I can repeat the feat in writing, without recourse to pictures or spreadsheets, and in much plainer language than Wikipedia.
It goes something like this:
Monty Hall is a game show host and you are the winning contestant. He shows you three doors and tells you that behind one is a car; behind the other two doors there are goats. You have to pick a door, and whatever is behind your choice is yours to keep.
Simple so far.
So you choose a door (say, the middle one), but instead of revealing what is behind it, Monty Hall opens one of the other doors - there is a goat munching some hay. He then asks, 'Before I open the remaining doors would you like to alter your choice?'
The Monty Hall problem is this - should you stick with your original choice or not?
Most people say it makes no difference: you have a 50:50 chance because there are two doors left - altering your choice would not change the probability of winning. Intuitively this seems right.
But most people are wrong! The correct answer is that you should switch your choice to the other door. In fact, you are twice as likely to win the car if you do so.
Can you figure out why?
The key to understanding the correct answer is to recognise that when Monty Hall opened the first door (to reveal a goat), he did not necessarily open it at random. This makes all the difference to the probability of winning if you switch.
Let's go through an example to see why.
Remember, there are three doors you can choose from - I'll call them A, B and C. Behind one door is a car, behind the others are goats.
We are going to assume the car is behind door A. Of course, you don't know that when you make your choice - but remember that Monty Hall does!
Now let's work through each of the options and see how switching your choice later on affects the outcome.
Remember, the car is behind door A.
Option 1 - you chose door A.
If you chose door A, Monty Hall can open any of the other doors at random (either B or C it makes no difference) and reveal a goat.
He then asks, do you want to stick with your choice of A or change?
Obviously, if you stick with your choice (A) you will win; if you switch (to B or C) you will lose.
Score so far: Sticking 1 - 0 Changing
Option 2 - you choose door B.
Now in this case - and this is the critical bit - Monty Hall can't open one of the other doors at random. He knows that the car is behind door A and you have chosen B, therefore he MUST open door C.
He then asks, do you want to stick with your choice of B or change?
If you stick with your choice (B) you will lose; if you switch (to A) you will win.
Score so far: Sticking 1 - 1 Changing
Option 3 - you choose door C.
This is exactly the same as the option above. In choosing door C you have picked a goat, so Monty can't open the other door at random - he MUST open door B otherwise he would reveal the car, which is behind A.
He then asks, do you want to stick with your choice of C or change?
If you stick with your choice (C) you will lose; if you switch (to A) you will win.
Score so far: Sticking 1 - 2 Changing
The wrap up
There were only three choices you could make - A B or C. The example above has covered all the permutations.
It shows that the person who sticks with their choice will win one out of three times - the person who switches will win two out of three times.
And the reason is that in two of the three cases (the times you will win by switching), Monty Hall is not choosing a door at random when he reveals a goat - he is choosing the only door that will keep the car concealed.
Putting this another way - if your initial choice was the door with the car, Monty can open any other door and you will lose if you switch. But if your initial choice was a goat, he must always show you the other goat and leave the car concealed. As your'e twice as likely to pick a goat with your first choice of door, you are therefore twice as likely to win the car if you switch.
So there you have it - the Monty Hall problem explained!