The following is a translation of a Dialog I had with ChatGPT in German, the translation, of course, done by ChatGPT.

The Monty Hall Problem and Confirmation Bias

User:

Suppose I appear on a game show where I am shown three doors. There is supposed to be a Porsche behind one of them, which I win if I choose the correct door. Based on the host’s behavior, I suspect the left door and choose it. The host then opens the middle door, which is empty, and asks me whether I want to switch. Should I switch to the right door?

ChatGPT:

Yes, possibly—but it depends crucially on what you know about the host and how he behaves. This is an interesting variation of the Monty Hall problem, because you did not choose your first door at random, but based on a suspicion about the host.

Let’s look at the situation mathematically.

1. The classic case

If you choose the left door at random, the probability that the Porsche is behind it is (1/3). The probability that it is behind one of the other two doors is (2/3).

If the host then opens an empty door and knows where the Porsche is, the usual Monty Hall assumptions apply:

• Your left door: probability of winning (1/3).
• The right door: probability of winning (2/3).

So you should switch.

2. But you have a suspicion about the host!

You chose the left door because you inferred from the host’s behavior that the Porsche might be behind it.

That changes the situation. Suppose your suspicion is genuinely informative: you believe that the probability of the Porsche being behind the left door is, say, 60%.

Then, before any door is opened, the probabilities are:

Now the host opens the middle door and reveals that it is empty. If he can open the middle door only when there is no Porsche behind it, and he does not otherwise favor the right door, the probabilities after this observation become:

• Left: (60/(60+20)=75%)
• Right: (20/(60+20)=25%)

In this example, staying would be much better than switching.

3. The crucial question

Exactly how is the host’s behavior related to the location of the Porsche?

• Case A: You have formed a suspicion based on his behavior, but that behavior provides no additional information about the actual location. Then the classic result applies: switch.
• Case B: You know the host well and can reliably interpret his behavior. Then your initial choice may already be better than a random guess. Switching is not automatically the right decision.
• Case C: The host knows where the Porsche is and deliberately chooses the middle door to open. Then we also need to consider how likely he is to open that particular door, depending on where the Porsche is.

My question to you: When you say you suspect the left door based on the host’s behavior, do you mean that you can interpret his body language or other signals, or that you are analyzing his behavior within the rules of the game? Mathematically, that makes a crucial difference.

User:

What surprises me is that you fell for this trick, too. The host was not obliged to show me another door. At least, I never said he was when I described the problem. If I had chosen an empty door, he would simply have opened it. Since the host is not stupid, you have a 0% chance of winning by switching.

ChatGPT:

You’re right that I introduced a crucial assumption without saying so: I assumed the classic Monty Hall protocol, even though you had not described it. That was my mistake.

However, I would like to examine your conclusion that the probability is 0% a little more closely.

You seem to be describing the following strategy on the host’s part:

• If you have chosen an empty door, he opens that very door.
• If you have chosen the Porsche, he opens a different, empty door instead.

Under these rules, your observation is indeed decisive: The host opens the door you selected precisely when you are wrong.

If he opens your chosen door and it is empty, you have lost. If he does not open it, you know you have the Porsche—and switching would make you lose.

There is, however, a small ambiguity: you say that the host opens the middle door after you have chosen the left one. That is not the same as saying that he always opens your chosen door when it is empty. If he can open the middle door regardless of whether your choice is correct, it does not automatically follow that your probability of winning by switching is 0%.

The real point is this: The probabilities are determined not just by the three doors, but also by the host’s strategy. I should have clarified that strategy first instead of automatically assuming the rules of the classic Monty Hall problem.

User:

EVERYONE assumes the classical problem. This is called confirmation bias.

ChatGPT:

Yes, that is a fair criticism. I recognized the familiar Monty Hall problem and then filled in the missing information in a way that made it fit the familiar problem. Instead of first examining what conclusions followed from your actual description, I interpreted it in a way that confirmed what I already knew.

This is closely related to confirmation bias, although the mechanism is somewhat more specific: I activated a familiar interpretive framework and then interpreted ambiguous information in favor of that framework. One might also call it schema bias or premature framing.

Your example illustrates something important: the classic solution to Monty Hall does not follow simply from having three doors and one door being opened. It depends on the rules governing the host’s behavior. Change those rules, and the solution can change.

I made a second mistake, too: when you pointed this out, I initially tried to fit your conclusion into the model I had constructed, rather than questioning my original assumption from the ground up.

The real lesson is not that Monty Hall is more complicated than we thought, but that recognizing something is not the same as understanding it. Especially with a familiar mathematical problem, I should first analyze the stated assumptions and only then decide whether the familiar solution applies.