If you are not already familiar with the Monty Hall Problem, here it is:
It appears like this is a fraud. I find it difficult to believe how it became so popular.
The confusion in the argument is that it doesn't consider the dynamic of time.
The moment something changes effecting the probability of the outcome, the entire decision should be reconsidered.
While it is true that at the time when the initial choice was presented and the first door was chosen, the odds of that door belonging to the car is 1/3. However, the moment the host opened a door to reveal a goat, the original decision must be reconsidered as the odds of the door chosen hiding the car are now 1/2. Therefore, it should make absolutely no difference if you remain with your original decision (which now has a 1/2 chance of belonging to the car) or switch (which also has a 1/2 chance of hiding the car).
Why do so many people fall for it? Can someone please explain?
As per Eli's suggestion (in the comments) I experimented with this here: http://www.nytimes.com/2008/04/08/science/08monty.html
here are my results:
but I can't see the programming, so it's not really a perfect experiment.
Here's the explanation provided there, for the first time it's beginning to make some sense:
"If your strategy is to always switch doors, you will lose only if your initial choice is the door with the car, which is a 33.3 percent chance. In the other two cases (66.7 percent of the time) you will switch to the car and walk away a winner."
In other words, sequence makes all the difference. After you have made a 1/3 chance choice, the game show host points you closer toward the more likely option by showing you which of the remaining two it is not.
Why do the odds of "not switching" not change now that one of the doors hiding the goats has been revealed? Why is it that if someone were to come and make a fresh choice with one door open revealing a goat and two doors closed one hiding a car and one hiding a goat, the options would be equal?
Consider it from this perspective: The game show host had to open a door revealing a goat. There is a 2/3 chance that you limited his choice to one door (if you chose a door with either of the two goats you would be forcing him to reveal the only other door with a goat), so there is a 2/3 chance that he revealed the other door with a goat, making it a 1/3 chance that you win if you stay with your choice but a 2/3 chance to win the car if you switch.
If you are presented with a first choice with one door open already revealing a goat, then your chances are 50/50, since there is no 2/3 chance that you forced the host to open the door with that goat and he could have just as well opened the door with the other goat.
