Instinct versus Reason

There used to be an American game show called 'Let's Make a Deal', hosted by a presenter called Monty Hall. Before you start thinking this is just an American version of 'Deal or No Deal', which isn’t my favourite show as you know, this one is different. Try and stick with me because this may not be easy at this time in the morning – and to give you an incentive, I will give a prize of £10 to the pupil who, by this time next week, can write to me with the best original explanation of what I am about to run through: the so-called ‘Monty Hall problem’. In case you want to read this again before responding, I will put this talk on the website.

Imagine you are a contestant on ‘Let’s Make a Deal’. The format of the show has three stages. 1: Monty Hall shows you three identical doors and tells you that behind one of the doors is a valuable prize, say a dream holiday, or a car. Behind the other two doors there is nothing. You have to choose the right door – if you do, you win the prize. Stage two: after you have chosen which door you want, but before the door is opened, Monty (who knows which door the prize is hiding behind) proceeds to open one of the remaining two doors and shows you that there is nothing behind it. The suspense rises, precisely because he always chooses a door which doesn’t have the prize behind it.

Finally, stage 3: and here’s the catch: Monty then gives you the option of sticking with your original choice or choosing to swap to the remaining unopened door. What should you do?

The ‘common sense’ answer seems to be that since Monty has shown you what’s behind one of the three doors, there are only two left. Each of these must be equally likely to be the winning door, so there is no reason to change your choice. You might as well stick with your original choice; you have a fifty-fifty chance. Sadly, whilst this answer seems instinctively appealing, it is in fact wrong. Probability theory dictates that you should choose to swap to the remaining unopened door – this will double your chances of winning.

This may be far from obvious to you and, in case you don’t see why, don’t worry, you are in good company. Mathematicians have almost come to blows in discussing this problem. Since the issue was first raised, it has generated huge controversy: there have been heated discussions via learned mathematical journals and even a couple of books that address the problem.

Let me try to explain why it makes sense to swap to the remaining unopened door. There are crucial two facts about Monty here, which change all the probabilities involved: 1) he knows where the prize is; and 2) he always chooses to open a door which has no prize behind it.

Bearing these two facts in mind, let’s go back to the start of the process. When you first guess which door you want, you have a one in three chance of guessing the door with the prize behind it. There is a two-thirds chance that you will have chosen one of the two prize-less doors. Then Monty opens his door, always being careful to choose a door which does not have the prize behind it. Remember that in two thirds of cases, you will have chosen one of the two prize-less doors. This means that when it comes to Monty’s choice, well, in two-thirds of cases, there is only one remaining prize-less door, so he has to open that one. If so, then the third door, the door which you didn’t originally choose and which Monty didn’t then open, will (in two thirds of cases) be the one with the prize behind it. So you should swap to that door. You won’t always win, but you will be twice as likely to do so than if you stick to your original choice.

What does this tell us? Well, it demonstrates that, whilst probability theory is a branch of mathematics for which most people have an instinctive or intuitive feel, unfortunately, our instinct is often wrong and we cannot always rely on it. Even in even fairly simple situations involving chance, you cannot entirely trust your intuitions. So whilst instinct is an important factor in decision-making, it isn’t foolproof. Reasoning combined with instinct will produce a better decision than one based on instinct alone.