In a certain game show, a contestant is presented with three doors. Behind one of the doors is a car, behind the others are goats. The contestant is asked to choose a door. The game show host, Monty, then opens one of the other doors to reveal a goat behind it. The contestant is then asked if she/he would like to stick with the original door or switch to the remaining door.
The contestant chooses Door 2. Monty opens Door 3 to reveal a goat. Monty knows which door has the car and always opens one that has a GOAT behind it. The dilemma is this.
Should the contestant:
Stick with Door 2?
Switch to Door 1?
Does it matter, i.e., could you flip a coin to decide?
Yes, there was a 2/3 probability that you picked a goat the first time. But that was based on all of the doors being closed. Change the circumstances, you change the probabilities. The wave function collapses not when you make the choice, but when the door is opened, and as conditions change, probability shifts. With one door opened and revealed to have a goat, the chance is now 50-50 either way.
This problem is interesting because the intuitive answer above is false. The important aspect Nav missed is that Monty always picks a goat to show. He knows where it is. So when a goat is revealed it has told us nothing about our original choice: its still 1/3 chance of being a car. But we have learned something about the other two choices: the 2/3 probability (of them together) has been placed on one door based on our new infomation.
Consider each door in turn. Say #3 has the car and the others are goats:
-If we pick door #1, we pick a goat. Then the other goat is revealed. If we switch, we get the car, if not we get a goat.
-Door #2 is the same. Switching gets the car, staying gets a goat.
-If we pick door #3, one of the two goats will be revealed, and if we switch we get the other.
These three scenarios exhaust the possibilities. And 2 of them favor switching, one does not. There is therefor a 2/3 chance of switching being beneficial. So, the odds favor switching.
. Try it yourself first.