Monty Hall
From Wikipedia:
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. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?
The answer as it is well known is that you should ALWAYS switch, a solution that most people(even mathematically sophisticated people) find paradoxical. But why do we find it paradoxical? Perhaps we humans have problems with conditional probabilities or maybe the phrasing of the problem introduces hidden biases. I think both of these are contributing factors.
I'd seen this problem several times but hadn't given it too much thought, but just the other day I saw an explanation that seemed to make things much clearer.
Imagine that instead of three doors three were 1,000,000. You choose a door and then Monty opens up 999,998 other doors, revealing goats. NOW should you switch?
I think this immediately reveals that the fact that Monty is ONLY going to open a door with a goat is quite significant. So there is a slight "magic trick" in the original statement as the crucial fact that only a door with a goat will be opened is easy to overlook when there are only 3 doors.
I suspect that some part of our brain thinks that Monty is opening a door "uniformly at random" and that is why we expect that there should be equal probabilities of getting a prize behind each door. Indeed consider the following variant
Imagine that Monty UNIFORMLY AT RANDOM "disappears" one door after you make your pick (this door may or may not contain the prize). Now you are given the choice to change your selection. Does switching give you any advantage?
This variant has the boring and expected solution that there is no advantage and this is probably what our brain is expecting in the original problem.
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. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?
The answer as it is well known is that you should ALWAYS switch, a solution that most people(even mathematically sophisticated people) find paradoxical. But why do we find it paradoxical? Perhaps we humans have problems with conditional probabilities or maybe the phrasing of the problem introduces hidden biases. I think both of these are contributing factors.
I'd seen this problem several times but hadn't given it too much thought, but just the other day I saw an explanation that seemed to make things much clearer.
Imagine that instead of three doors three were 1,000,000. You choose a door and then Monty opens up 999,998 other doors, revealing goats. NOW should you switch?
I think this immediately reveals that the fact that Monty is ONLY going to open a door with a goat is quite significant. So there is a slight "magic trick" in the original statement as the crucial fact that only a door with a goat will be opened is easy to overlook when there are only 3 doors.
I suspect that some part of our brain thinks that Monty is opening a door "uniformly at random" and that is why we expect that there should be equal probabilities of getting a prize behind each door. Indeed consider the following variant
Imagine that Monty UNIFORMLY AT RANDOM "disappears" one door after you make your pick (this door may or may not contain the prize). Now you are given the choice to change your selection. Does switching give you any advantage?
This variant has the boring and expected solution that there is no advantage and this is probably what our brain is expecting in the original problem.