The donkey and the money

You are participating in a TV quiz and you have reached the final round. You have to choose between three doors: A, B and C. Behind one door there is a cheque of 5 million euro. It will be yours if you choose that door. If you choose one of the other doors you’ll win a donkey. You love donkeys, but you prefer the money, also because you can buy then many monkeys plus you’ll have enough money for taking care of the donkey. So you want to win the 5 mln euro but you have no idea behind which door the cheque might be. The quizmaster doesn’t give you a hint. At last you choose Door A. “Okay”, the quizmaster says, “are you sure?” “Yes, I am”, you reply. “Then I’ll open one of the other doors. I know behind which door the money cheque is and I’ll open a door with a donkey”, so the quizmaster. He opens Door C. You see a donkey. “Dear Harry”, the quizmaster then says. “You have chosen Door A. However, the money might also be behind Door B. As I told you, I know behind which door the cheque is. Do you want to change your choice or do you still stick to Door A?” You are a rational man, or so you think: “There are two doors. The cheque is behind one door and behind the other one there is a donkey. So, the chances are even that the cheque is either behind Door A or behind Door B. It makes no difference which door I’ll choose. So why change? It has no sense”. You stick to A. You are lucky: The quizmaster opens Door A and you see the cheque.

Now you are a rich man, a millionaire, for you have won 5 mln euro. You are a donkey lover, so you’ll buy a donkey for the money you got. But was it rational to stick to your choice of A, because the chances that the cheque was either behind Door A or behind B were even? Most people will say it was. If they would have been in your shoes in the quiz, they would have thought the same and there is a good chance that they had stuck to their choice, too; for psychological reasons (but that’s another story). However, they and you are not right. It would have been rational to change your choice to B. Let me explain.

There are six possibilities how the money cheque and the donkeys are divided over the doors. I have written them out in a table:

	Door A	Door B	Door C	win/loose
1	5 mln	donkey A	donkey B	−
2	5 mln	donkey B	donkey A	−
3	donkey A	5 mln	donkey B	+
4	donkey B	5 mln	donkey A	+
5	donkey A	donkey B	5 mln	+
6	donkey B	donkey A	5 mln	+

Let’s suppose that you have chosen A and the quizmaster opens a door with a donkey behind it. Then you change your choice to B or to C, as the case may be. The last column of the table shows what happens. If division 1 is the case, you are out of luck: The cheque is behind Door A and you have changed to a door with a donkey. Therefore I have written a minus sign in the last column. Also in situation 2 you are out of luck and will get a donkey. But in the situations 3, 4, 5  and 6 you’ll change to the door with the cheque, since the quizmaster has opened already the only door with the monkey. So the odds are two to one that you’ll win the cheque, on condition that the quizmaster knows behind which door the cheque is (and so opens the other one with a donkey).

But how about if you had stuck to your choice of Door A? Then you had won the money in situations 1 and 2 but you had got a donkey in all other situations (the minus signs become plus signs in the last column of the table and the other way round). Now the odds are one to two to get the cheque.

Was it rational to switch? Now you’ll say “yes”: It does sense to change your choice because the quizmaster knows what he does, when he opens one of the doors you hadn’t chosen. But most likely you’ll not be the only person who makes this mistake, unless he or she has read the explanation. Even more, after it had been published (in the American Statistician and elsewhere), still many readers thought that the chances were even. Among them there were highly educated and knowledgeable people. Rationality is often not a matter of knowing the right thing but a matter of psychology. Know who you are and what rationality means.

Source: Herman de Regt & Hans Dooremalen, Het snapgevoel. Amsterdam: Boom, 2015; chapter 5. If you want to know more about it, google then “Monty Hall problem”.