No, not the goat problem again. Yes, the goat problem again. I can still remember the first time I encountered this problem. My highly esteemed fellow programmer Thomas Bez, greetings Thomas, brought me an article like this, in 1990, and seemed very excited. It said that there was a quiz game in America. And in this game the following situation arose. The contestant saw three locked doors in front of him. Behind one of these doors was a car, and behind each of the other two doors was a goat.
The candidate was allowed to choose one of the doors, but was not supposed to open it immediately. Well, we, as practised probability calculators, divided one favourable door by three possible doors and got a 1/3 chance of winning the main prize. That simply has to be right. Then the following interlude: The quizmaster opened one of the other two doors. One behind which there was a goat. Then he asked the candidate if he wanted to stick with his choice or perhaps switch to the remaining third door, which was also still locked. Well, most of the candidates politely thanked him for the nice offer, but did not want to change their first choice. Why should they? They now had one of two doors to choose from. Either they now had the right one or the wrong one, fifty-fifty I guess you say.
So my esteemed colleague showed me all this. And got excited. Because: a hypercritical reader would have suggested to the candidates that they should rather change the door. They would increase their chances of winning, even double them!
It was not even enough to get upset about such nonsense, but in this case a letter to the editor had to be written. It had been written and sent long ago. I heard that the magazine had received thousands of letters with similar wording. How it could be possible that such a popular and scientifically recognised newspaper could publish such nonsense. The chance is fifty-fifty, no more and no less.
The magazine even went so far as to make the outrageous claim that computer programmes had been developed to find out in a simulation what the actual distribution of chances was! And these would have delivered astonishing results: they would have supported the lady’s claim.
Personally, I even read quite a few of the letters to the editor later, as some of them were published. The content was comparatively less interesting than the strange clustering of the signatories of doctors and professors. Thomas had even asked for the programme code in order to uncover the error in the code!
This was a little strange for me. I asked Thomas to take a deep breath and sit down. Then I explained to him, gently and as gently as possible, that — the lady was right.
The ensuing discussion did not shatter any friendship for only one reason: I had taken the precaution of getting a pack of cards. I took out an ace and two other random cards. Paper and pencil were also ready. “Please, dear Thomas, I am the contestant, you are the quizmaster. I choose a card, then you turn over one of the other two cards. Then I get to decide whether to switch. Then you write down whether I found the ace in this attempt or not. At the end, we’ll count the right and wrong ones together. Agreed?” “Agreed.” So we began the experiment. I had no intention whatsoever of doing the game for money, that much by the way. After about 20 tries, he stopped the experiment. And it wasn’t just luck that I had such a high hit rate in those 20 attempts. He simply noticed while doing the experiment that every time I didn’t choose the ace in my first choice, he always had to think briefly about where it was and then he had to turn over the other card.
The answer is now really clear: you have to switch.
But there are still some aspects that need to be explained, ways of looking at how the confusion came about and even ways of looking at the problem itself.
First of all, it is astonishing that such confusion can be created with a simple experimental set-up. Then the intuitive reaction of most people to whom the problem is first described. And then the fact that it is enough to cause a frenzy, i.e. that highly educated people can get carried away with a phantom idea. And, last but not least, that such a simple experiment, which anyone could carry out immediately and anywhere, would be sufficient to resolve the confusion. At least, after carrying out the experiment, one would have confirmation for oneself that one had made a mistake in thinking. This would not solve the problem, but at least one could save oneself the embarrassment of announcing one’s (wrong) assessment. And I am also convinced that anyone who carries out the experiment will soon realise where the error in thinking lies.
To discuss the problem itself, I consider, for example, that the candidate obviously has a 33.33% chance of being elected the first time round. And the interlude of the open door cannot increase this chance. I then imagine that the candidate chooses a door. Now he has his 1/3 chance. No one would doubt it. Then the quizmaster says he will now open a door. The candidate who is unwilling to change says: “Please, open, but I won’t look at what’s there, I don’t want to know. I’ll stay with the door.” The quizmaster nevertheless opens a door, according to his pre-set rules. Then the candidate opens his eyes again and, hocus pocus, he has 50%.
The observation that there are 100 doors is also often used. The candidate, completely unaware, goes to any door and stands in front of it. Then the quizmaster starts opening doors. Now the candidate should begin to rub his hands together. “You see, I didn’t guess too badly. My chances go up and up, now I’m already at 10%, 20%, 30%.” The quizmaster keeps opening new doors, the chances increase. The penultimate door is opened. Another goat! Then the contestant says to himself, “Well, then the car must be here where I’m standing.”
Or is it perhaps the only door not yet opened by the quizmaster?
Oh, I almost forgot. How do you actually work it out?
Well, if you stick to the three-door problem, the chance you get by switching is of course 2/3. The reasoning is simple: You have 1/3 with the door you are standing in front of. The remaining 2/3 are behind the other door. After all, it has to be somewhere.
There is a very last way of looking at it, which you only discover when you carry out the experiment, so to speak: If one has chosen with the immovable probability of 1/3 /with 100 choices 1/100) and it is the quizmaster’s turn, then he has two possibilities, two doors to choose from. 1/3 of the time there is nothing behind both doors. Because the candidate has found the goat, it is behind the chosen door. In the other 2/3 of the cases, behind one door is the goat, which the moderator could open. There is nothing behind the other door. Now he could just as well say: “Dear candidate. You have now chosen this door. You had a 1/3 chance. But now I will show you where the goat is actually located. Namely here.” And he could then open the door with the car and then ask if the candidate wants to change. The rules don’t allow that, so he shows the door where there is a goat behind it and says, in effect, with a wink and behind closed doors: “(I’m not allowed to open the door where the car is, only the other one, but now you’ll have to find out for yourself where it actually is. Well, right, logical, behind the other one.)”
Even easier with the 100 doors. You’ve chosen one of them, just the tiny chance of 1/100th, so very rarely the right one. Then it’s the moderator’s turn. And now instead of the other 98 doors where there are goats behind, he could refer directly to the “right” door. “Ok, you chose the door, that’s the wrong one (sure, almost always). Here is the right one.”
When conducting the experiment, it is just in the majority of cases (2/3 with 3 choices) that player 1 has not found the ace. Then it’s player 2’s turn and in these cases he says: “Ok, you’ve chosen the card, now I’ll show you which one is the right one”.