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While reading about the Monty Hall problem, I read that the probability of the car being behind the chosen door is 1:3. When Monty opens a door, the other door's probability of having the car goes from 1:3 to 2:3.

Even though this provides the right answer, is it correct to describe the problem in this way?

I would say that one door has a 100% probability of having the car behind it and the other two doors have a 0% chance of having the car. The fact that the contestant does not know where the car is does not change this. Am I wrong?

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Statements of probability always refer to some frame of reference and knowledge. The probability the car is behind a particular door is either 0% or 100% IN THE FRAME OF THE TV STAGE ASSISTANT WHO PLACED IT THERE. But IN THE FRAME OF THE CONTESTANT (at the beginning), the probability is 33.3% for each door. According to the Monty Hall scenario, after a guess and one of the doors is opened (revealing a goat), then the probabilities are 33.3% and 66.6% for the contestant. Nevertheless, the probabilities remain 0% and 100% for the TV stage assistant throughout.

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The Purpose of Probability

Probability helps us deal with uncertainty and so the way we express probabilities is dependent on the perspective that we are tackling a problem from.

For example, if the game host knows which door the car is behind, then there is no uncertainty and so the probability from the host’s perspective of the car being between that particular door is $100$%.

However, the purpose of probability is often to help us to make rational decisions (I would recommend looking into game theory, decision theory, and prospect theory for more on this).

Therefore, if we look from the contestant’s perspective, they would assess the probability of the car being behind door $1$ as being $1/3$ (if there are $3$ doors and $1$ car). Even though the car is behind only one door with certainty, they are using probability to assess their chances of success in order to increase their odds of winning the game.

The player can use their own individual assessment of the probability of the car being behind each door in order to form a winning strategy (we wouldn’t be able to build a winning strategy if we just considered the probability that it is behind exactly one of the doors to be $100$% because the contestant doesn’t know which door that is).

Adapted Monty Hall Problem

It is easier to see this if we look at an adaptation of the original problem. Consider the following scenario:

There are now $100$ doors and only $1$ door has a car behind it. You are asked to pick one door that you think the car is behind. Let’s assume you pick door $5$. Now every door is opened and the game is narrowed down to picking between your choice (door $5$) and door $72$. Should you switch?

Now it seems obvious that we should switch. However, in the original game if we applied the same probabilistic logic regarding conditional probabilities then we would arrive at the same answer (that we should switch in order to maximise our odds of success).

And so whilst you are correct that the car is behind one door with probability $100$%. That isn’t useful for us which is why we use probability to express our individual level of uncertainty in this situation to make good decisions. If the player just considered one door to have the car with $100$% certainty then this isn’t beneficial in terms of developing a good strategy to play by.

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