The Monty Hall Problem
Now this is beautiful mathematics. Here's the problem:
"You are on a game show on television. On this game show the idea is to win a car as a prize. The game show host shows you three doors. He says that there is a car behind one of the doors and that there are goats behind the other two. He asks you to pick a door. You pick a door but the door is not opened. Then the game show host opens one of the doors you didn't pick to show a goat (because he knows what is behind each door). Then he says that you have one final chance to change your mind before the doors are opened, and asks you if you want to change your mind and pick the other unopened door instead. What do you do?
Using your intuition you might think that there is a 50-50 chance that there is a car behind each door. But according to Marilyn vos Savant you should always change and pick the other door because chances are 2 in 3 that there will be a car behind that door. 92% of the readers wrote in to Marilyn and said she was wrong, many of whom have Ph.D's in mathematics."
But Marilyn is right, do you see why? If you're really curious and want to see why, I posted the solution here (look at all comments, but the final solution is in the last comment I posted).
Amazing, huh?
"You are on a game show on television. On this game show the idea is to win a car as a prize. The game show host shows you three doors. He says that there is a car behind one of the doors and that there are goats behind the other two. He asks you to pick a door. You pick a door but the door is not opened. Then the game show host opens one of the doors you didn't pick to show a goat (because he knows what is behind each door). Then he says that you have one final chance to change your mind before the doors are opened, and asks you if you want to change your mind and pick the other unopened door instead. What do you do?
Using your intuition you might think that there is a 50-50 chance that there is a car behind each door. But according to Marilyn vos Savant you should always change and pick the other door because chances are 2 in 3 that there will be a car behind that door. 92% of the readers wrote in to Marilyn and said she was wrong, many of whom have Ph.D's in mathematics."
But Marilyn is right, do you see why? If you're really curious and want to see why, I posted the solution here (look at all comments, but the final solution is in the last comment I posted).
Amazing, huh?
posted by Sina at
3:38 AM
3 Comments:
I have a joke...
There are 3 men on a train. one of them is an economist, one is a logician, one is a mathematician. And they have just crossed the border into Scotland, and they see a brown cow standing in a field from the window of the train and the cow is standing parallel to the train.
The economist says, "look, the cows in Scotland are brown."
The logician says, "no. There are cows in Scotland of which one at least is brown."
And the mathematician says, "no. there is at least one cow in Scotland, of which one side appears to be brown"
By
Fashionista, at December 19, 2005 10:25 PM
Ok mister, you got me intrigued, and I come looking for the solution...
Where is it? Why is she right? I don't get it, and want to.
By
Andy Hunter, at January 30, 2006 1:02 AM
So you have 3 doors. You randomly pick one. The game show host shows a door with a goat, and then gives you the chance to switch doors...blah blah.
Now, suppose you decide to ALWAYS switch doors after the host shows you a goat. Think about it. Remember you will ALWAYS switch (so there's no deciding really). You'll have two cases:
1) You choose a door with a goat behind it. The host shows you the other goat. You switch doors -> you get the car.
2) You choose the door with the car behind it. The host shows you either door with a goat. You switch doors -> you get a goat
Assuming you're going to switch, AS LONG AS YOU PICK A DOOR WITH A GOAT FIRST YOU'LL GET THE CAR!! But what is the probability of initially picking a goat? Well, 2 of 3 doors. So 2/3.
Thus, if you switch your door, the probability of getting the car is 2/3.
Does that make sense?
By
Sina, at January 30, 2006 11:56 PM
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