The Monty Hall Paradox Explained
A simple explanation of a complex problem.
The "Monty Hall" paradox is one of the most widely misunderstood probability problems in existence. The problem, even when explained without any ambiguity, confuses even many scholars.
The problem goes like this. You are a contestant on Let's Make a Deal, an old game show hosted by Monty Hall. You are given a free choice of three doors. Behind one of the doors is a new car, and behind the other two are goats. After you have made your selection, Monty opens one of the doors that you did not choose, revealing a goat (Note: Monty knows what is behind each door and will always reveal a goat). He then offers you the choice of sticking with your original choice, or trading it for the remaining door. Is it better to keep your initial choice or to switch, or are your chances of getting the car the same either way?
The majority of people intuitively state that the odds of winning the car are 50/50, regardless of whether the contestant keeps their original choice or not. This is, however, not the case. Switching to the other door doubles the probability of winning the car, a result that seems completely absurd to most people.
When you make your initial selection, there is a 1/3 chance (33.3%) that you have chosen the door concealing the car. There is therefore a 2/3 chance (66.7%) that the car is behind one of the other two doors. If Monty then removes one of the other two doors, your original choice still only has a 1/3 chance of concealing the car, leaving the other door with a 2/3 chance of concealing the new car.
Even when this concept is explained, many people still have trouble believing that the result is true. It is easy to prove by demonstrating all possible scenarios, though. For the purposes of our demonstration, we will place the car behind Door #3.
| Door #1 | Door #2 | Door #3 |
| Goat | Goat | Car |
- If you initially select Door #1, then Monty will open #2 to reveal a goat. Keeping your choice results in a goat, while switching results in the car.
- If you initially select Door #2, then Monty will open #1 to reveal a goat. Keeping your choice results in a goat, while switching results in the car.
- If you initially select Door #3, then Monty will open either #1 or #2 to reveal a goat. Keeping your choice results in the car, while switching results in a goat.
This example is the same no matter which door initially conceals the car. Two times out of three, switching doors will result in winning the car.
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Comments
hey nice article. This is definitely hard to understand and you made it a little easier to do so.
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