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Conditional Probability: The Monte Hall Problem

From Theory To Practice

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The Monte Hall Problem is a classic probability puzzle, where a game show contestant is trying to find the car from behind three possible doors, each of which has either a goat or the car behind it. After the contestant randomly chooses a door, Monte Hall (the game show host) eliminates one of the other two doors that does not have the car. The contestant is then asked if he would like to switch his choice, and while it appears to be a 50/50 proposition, the contestant would always be better off switching.

The initial choice by the contestant has exactly a 1/3 probability of having the car, while there exists a 2/3 probability that the car is behind one of the other two doors. In order to eliminate one of the other two doors, Monte has to “filter” these choices. While searching through these doors, if he finds the car, he must leave it, and instead filter out the other door that contains a goat. In other words, Monte will never filter out the car because somebody has to win the game, and if he did, then the game would come to an abrupt stop. That can obviously never happen. Thus, the contestant is always better off switching because there is a 2/3 probability that Monte will filter out a goat, and he will leave you with the car behind the “other” door.

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