Monty Hall Problem and Conditional Probability

Monty Hall Problem and Conditional Probability

The Monty Hall Problem is a thought experiment in which one asks, "If I choose a door in a game and the host opens the wrong door, should I change my choice?" This problem is often introduced as a representative example of so-called conditional probability, in which probability changes depending on the information one has.

The rules of the game are as follows: there are three doors, one of which hides a prize (e.g., a car), and the other two hide a goat (a mislay). First you choose one of the doors at random (at this point the chance of winning is 1/3). Next, the host (Monty Hall) opens one of the other two doors you did not choose, which must have a goat. You are then asked, "Would you like to change to the other remaining door?" You will be asked - and so on.

Intuitively, you might think that the two remaining doors would be 50/50 (1/2/1/2), but this is incorrect. The probability that the winner was chosen at the beginning was 1/3, and the probability that the winner was chosen for the wrong door was 2/3. Now that the moderator has flushed out the hazard, all the "non-winning possibilities" of the 2/3 that had chosen the hazard are concentrated in one other door that remains - this change in information changes the likelihood of the door being a winner. As a result, we can conclude that the probability of winning remains 1/3 if we "leave the door as it is", but there is a 2/3 chance of winning if we "change the door", i.e., changing the door "doubles the probability of winning".

Thus, what the Monty Hall problem shows is that probabilities change depending on the underlying information structure (who knows and which doors are excluded), even if the choices do not seem to "change". In other words, probability should be treated not as a "mere number" but as a "conditional belief" that reflects "which information one has" - that is the essence of conditional probability. This is the essence of conditional probability, and why it is so important in the world of "informed judgment" in areas such as competition, risk assessment, cybersecurity, and decision making.