Quiz:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
The essence of the improved probability is in the host's prior knowledge of where the car is. If he opened the door randomly, your probability would not change.
This always puzzled me too, because your odds are better picking the other door, but if someone just came on the scene, his are lower, so you have inside information.
The players who swap have a 2/3 chance of winning the car and players who stick have a 1/3 chance of winning the car, is based on the premise that the host knows which door hides the car and intentionally reveals a goat. If the player selected the door hiding the car (1/3), then both remaining doors hide goats and the host may choose either door at random, and switching doors loses. On the other hand, if the player initially selected a door that hides a goat (a 2-in-3 chance), then the host's choice is no longer at random, as he has no options but to show the other goat, the second goat, and switching doors wins for sure.
Someone that disagrees
""You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you’ve chosen, we’ve learned nothing to allow us to revise the odds on the shell under your finger."
Back to the thread, don't want to hijack.