After the last Dronfield Branch Labour Party meeting several of us retired to the bar for a chat. I was keen to get their views on a puzzle my son-in-law had recently come across:
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 1, and the host, who knows what's behind the doors, opens another door, say 3, which has a goat. He says to you, "Do you want to pick door 2?" Is it to your advantage to switch your choice of doors?
According to this article you should always switch. I agree, but I had great difficulty convincing the others.
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 1, and the host, who knows what's behind the doors, opens another door, say 3, which has a goat. He says to you, "Do you want to pick door 2?" Is it to your advantage to switch your choice of doors?
According to this article you should always switch. I agree, but I had great difficulty convincing the others.
Very profile !
ReplyDeletegood post's
Byeee!
I received an email from Harry finally agreeing with me! He said...
ReplyDeleteThe reason that it pays to move away from your first choice to the second remaining door, arises from the actions of the doorkeeper.
Each time you initially choose a door from the three that are available, then you have a 3 in 1 chance of being correct. When the doorkeeper then opens a remaining door (and leaves only two to choose from) something unexpected happens. The first door you opted for does not have its odds changed by the doorkeeper's actions, so that it is still likely to be the correct in one case out of three. But the odds for the remaining door has been cut from "3 to 1" to "3 to 2" because one of the non-prize giving doors have been removed.
So if someone conducts a large enough set of experiments by using the doors (or cards or oranges), then the first options will be correct 1 in 3 times, but the other door will be correct 2 in 3 times. This is because the doorkeeper keeps taking out a door which is never the correct one (so it as is it were correct zero times out of all the 3s).
At first sight this appears to be illogical, because when there are only two doors left to choose from you would expect the odds on finding the correct one to be 2 to 1. But this to ignored the fact that a two stage selection processes has been undertaken. The doorkeepers actions don't change the 3 to 1 odds on your first choice, but he then selects between the other two doors when he opens one (which scraps its odds altogether) and transfers those odds to the remaining door which you did not initially opt for.
This is hard to perceive, because it is rather like trying to look into a four dimensional universe when your mind is attuned to a three dimensional one. No wonder some Maths Professionals with doctorates disputed the claim of the women who came up with this. And worse still even someone with a degree in Philosophy (including logic) could argue the point against you!
I woke up at 4am, couldn't get back to sleep because the problem kept going around in my head. Then when I thought I had cracked it I got up and drafted the above response. I then managed to get back to sleep at 5am.
Without undertaking my own experiments, I accept what you say as a matter of pure logic. I think.
Sweet dreams,
Harry.