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In the Monty Hall problem, why can you just assume the contestant picked door 1? Why are you entitled to relabel the doors, or rewrite this solution with the door numbers permuted?

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My bafflement ought be obvious. 1. A contestant could've picked doors 2, 3. So you can't just assume he picked door 1.

1. Correct me if I'm wrong, but the game show didn't authorize contestants "to relabel the doors, or" permute the door numbers. So what permits you to do any of this in this solution?

Example 2.7.1 (Monty Hall).

On the game show Let's Make a Deal, hosted by Monty Hall, a contestant chooses one of three closed doors, two of which have a goat behind them and one of which has a car. Monty, who knows where the car is, then opens one of the two remaining doors. The door he opens always has a goat behind it (he never reveals the car!). If he has a choice, then he picks a door at random with equal probabilities. Monty then offers the contestant the option of switching to the other unopened door. If the contestant's goal is to get the car, should she switch doors?

Blitzstein. Introduction to Probability (2019 2 ed). pp 68-9.

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Correct me if I'm wrong, but the game show didn't authorize contestants "to relabel the doors, or" permute the door numbers. So what permits you to do any of this in this solution?

The game show never labeled the doors in the first place. We are the ones labeling the doors, so we are free to label them as we please.

It might be more helpful if you just think of it as something more like "After the contestant picks a door, we label that door 'door 1'." Then we aren't relabeling anything, just flat out saying that no matter which door they choose, we call it door 1.

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