22.25 The Monty Hall problem

Probability is an important idea in science. An experiment to answer a question like, for example, whether a drug is safe can only be answered with a probability (post 16.1). When we measure anything we only really know that its value lies between certain limits with a given probability (posts 16.7, 16.24, 16.26, 16.28). Probability is central to the concept of entropy (posts 16.38, 19.30) and to the usual interpretations of quantum mechanics (posts 19.26, 19.27, 19.28, 19.29).

So how well do you think you understand probability? To find out, try solving the Monty Hall problem. This problem is named after the Canadian television host Monty Hall (1921-2017) who lived and worked in the USA.

There are three boxes – one contains a prize, the other two contain something worthless. But you don’t know which box contains the prize. It seems to be conventional to describe the prize as a car and the other things as goats. Really they should be called tokens to win a car or a goat because you could hear which boxes contained real goats. Also, if you think a goat is a valuable prize, pretend you would prefer the car!

In Monty Hall’s television programme, you choose a box but it stays closed. Monty then opens another box that he knows is a goat box.

He then asks you if you want to stick with your original box or switch to another. For example, if you chose box 1 you can stick with this box or switch to box 2 or box 3. You won’t want to switch to box 2 because that will be the box he chose to open and that you now know is a goat box. So your choice is really to stick with box 1 or switch to box 3.

What do you do? And does it make any difference?

Think about it for a while before you look at the appendix

Keep thinking!

Think a while longer!

Appendix

You are more likely to win the car if you switch your original choice!

Let’s suppose your first choice was box 1. If you stick with this choice, you will win a goat. If you switch, you will not pick box 2 – because you now know that you would win a goat. So, you will choose box 3 and win a car.

If your first choice was box 2 and you stick with your choice, you will win a goat. But you now have been shown that if you choose box 1, you will win a goat. So, you choose box 3 and win a car.

If your first choice was box 3 and you stick with your first choice, you will win a car. If you switch, you will choose box 1 or box 2 – bad luck, you win a goat!

These results are shown in the picture below.

If you stick with your original choice the probability of winning a car is 1/3. If you switch your choice, the probability of winning a car is 2/3.

What’s happening here? When you made your original choice, the probability of winning a car was 1/3. If you stick with that choice, the probability that you have won the car remains 1/3. But if you switch your choice, you can make use of extra information and so increase your probability of winning.

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