Apathy

10-15-2008, 09:36 PM

As long as these threads are so popular now, I figured I'd make one on pretty much one of the most famous math incorporated riddles in the world.

Skip down for the riddle, this is just my rambling because I'm long winded I guess.

It's one of Marilyn vos Savants' for those of you who don't know, she's the holder of the world's highest IQ, or at least she was, and I doubt anyone overtook her since. (She's also married to the guy who invented the artificial heart, who ironically enough, had to go to an italian medical school because he wasn't admitted to any others, in a small part because of his surprisingly low IQ, in the 80's iirc.)

Anyway, riddle time. I first heard this in my stats class last year. It was printed in a column Marilyn writes in 1990, and when she published the answer, mathematicians and statisticians from around the country wrote in explaining to her why she was wrong, or to condemn her for trying to trick modern day Americans.

But she wasn't wrong.

RIDDLE

"Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, 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?"

Highlight for answer.

The wrong answer? No.

Any reasonable person will see that the door the host opened has a goat, and assume that door #1, and door #2 both have a 50% chance of having a car.

Unfortunately, this is incorrect. In my opinion, the easiest way to prove the correct answer, is to look at all of the possibilities.

DOOR 1 DOOR 2 DOOR 3 RESULT

GAME 1 AUTO GOAT GOAT Switch and you lose.

GAME 2 GOAT AUTO GOAT Switch and you win.

GAME 3 GOAT GOAT AUTO Switch and you win.

GAME 4 AUTO GOAT GOAT Stay and you win.

GAME 5 GOAT AUTO GOAT Stay and you lose.

GAME 6 GOAT GOAT AUTO Stay and you lose.

When you switch doors, you win 2/3 games. When you stay with what you have, you only win 1/3.

The reason this works is that The Host will always eliminate a door with a goat in it. He can do this because he knows what is behind the doors. So in game number three from above, You Pick Door #1. The Host is forced to eliminate Door 2.

Another way to look at it. When you make your first choice, before the host eliminates any, there is a 1/3 chance of you having the car, and a 2/3 chance of the other doors having it. If the host eliminates one of the other doors, the probability doesn't change, because there's still three doors. You just know one of them now, and know not to pick it. So the door he didn't pick gets all of that 2/3, and your door still has 1/3.

Fuck Yes.

Skip down for the riddle, this is just my rambling because I'm long winded I guess.

It's one of Marilyn vos Savants' for those of you who don't know, she's the holder of the world's highest IQ, or at least she was, and I doubt anyone overtook her since. (She's also married to the guy who invented the artificial heart, who ironically enough, had to go to an italian medical school because he wasn't admitted to any others, in a small part because of his surprisingly low IQ, in the 80's iirc.)

Anyway, riddle time. I first heard this in my stats class last year. It was printed in a column Marilyn writes in 1990, and when she published the answer, mathematicians and statisticians from around the country wrote in explaining to her why she was wrong, or to condemn her for trying to trick modern day Americans.

But she wasn't wrong.

RIDDLE

"Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, 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?"

Highlight for answer.

The wrong answer? No.

Any reasonable person will see that the door the host opened has a goat, and assume that door #1, and door #2 both have a 50% chance of having a car.

Unfortunately, this is incorrect. In my opinion, the easiest way to prove the correct answer, is to look at all of the possibilities.

DOOR 1 DOOR 2 DOOR 3 RESULT

GAME 1 AUTO GOAT GOAT Switch and you lose.

GAME 2 GOAT AUTO GOAT Switch and you win.

GAME 3 GOAT GOAT AUTO Switch and you win.

GAME 4 AUTO GOAT GOAT Stay and you win.

GAME 5 GOAT AUTO GOAT Stay and you lose.

GAME 6 GOAT GOAT AUTO Stay and you lose.

When you switch doors, you win 2/3 games. When you stay with what you have, you only win 1/3.

The reason this works is that The Host will always eliminate a door with a goat in it. He can do this because he knows what is behind the doors. So in game number three from above, You Pick Door #1. The Host is forced to eliminate Door 2.

Another way to look at it. When you make your first choice, before the host eliminates any, there is a 1/3 chance of you having the car, and a 2/3 chance of the other doors having it. If the host eliminates one of the other doors, the probability doesn't change, because there's still three doors. You just know one of them now, and know not to pick it. So the door he didn't pick gets all of that 2/3, and your door still has 1/3.

Fuck Yes.