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Lets us say there are three doors. one of the doors contain a car and other doors contain goat. You want to win the car. Lets say you choose first door. The host opens up third door which has goat. As you know, if you switch to second door there is 2/3 chance that you will the car. Lets say you switch to second door but didn't open it. Keep in mind that second door has 2/3 chance to win the car(we don't open it)

Now lets repeat the experiment but instead your first choice was the second door. Now switching to first door there is 2/3 chance to win the car.

Now if we keep the first and second experiment in mind (let us say the things behind the door hadn't changed) then first and second door has 2/3 chance to win the car? See let it be the same person and same host not different people and its just that experiment was repeated but there is a change in first choice and in first experiment the switched door wasn't opened. How is two doors having 2/3 chance?

hackerking
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1 Answers1

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The monty hall problem does not say

"this door has this and this probability of winning a car".

What the problem says is

"Having picked one door, and the host having opened one of the remaining doors, the chance of the car being behind the unpicked door is $\frac 2 3$.

So, what you are describing is not that the first and the second door both have a probability of $\frac23$. That would be impossible. It is rather that:

  1. If you pick the first door, and the host opens the third, then the probability of the second door having the car is $\frac23$.
  2. If you pick the second door, and the host opens the third, then the probability of the first door having the car is $\frac23$.

Unlike your interpretation, the options above are not mutually exclusive. They are, in fact, both true.

The mistake you are making is that you are adding up conditional probabilities without actually taking the condition they are based on into consideration. For example, say I listen to the following weather report:

  • There is a 50% chance that there will windy.
  • If it will be windy, there is a 90% chance of rain.
  • If it will not be windy, there is a 60% chance of rain.

Would you say that the forecast is wrong, because 90+60=150?

5xum
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