LESSWRONG
LW

All of logical's Comments + Replies

6TylerK
The thing you might be overlooking is that Monty does not open a door at random, he opens a door guaranteed to contain a goat. When I first heard this problem, I didn't get it until that was explicitly pointed out to me. If Monty opens a door at random (and the door could contain a car), then there is no causal link and therefore the probability would be as you describe.
7JoshuaZ
At this point you've had this explained to you multiple times. May I suggest that if you don't get it at this point, maybe be a bit of an empiricist and write a computer program to repeat the game many times and see what fraction switching wins? Or if you don't have the skill to do that (in which case learning to program should be on your list of things to learn how to do. It is very helpful and forces certain forms of careful thinking) play the game out with a friend in real life.
pjeby100

You got a 50% chance.

No, you don't. Switching gives you the right door 2 out of 3 times. Long before reading this article, I was convinced by a program somebody wrote that actually simulates it by counting up how many times you would win or lose in that situation... and it comes out that you win by switching, 2 out of 3 times.

So, the interesting question at that point is, why does it work 2 out of 3 times?

And so now, you have an opportunity to learn another reason why your intuition about probabilities is wrong. It's not just the lack of "memory" that makes probabilities weird. ;-)

Reply
3Sideways
Your analogy doesn't hold, because each spin of the roulette wheel is a separate trial, while choosing a door and then having the option to choose another are causally linked. If you've really thought about XiXiDu's analogies and they haven't helped, here's another; this is the one that made it obvious to me. Omega transmutes a single grain of sand in a sandbag into a diamond, then pours the sand equally into three buckets. You choose one bucket for yourself. Omega then pours the sand from one of his two buckets into the other one, throws away the empty bucket, and offers to let you trade buckets. Each bucket analogizes to a door that you may choose; the sand analogizes to probability mass. Seen this way, it's clear that what you want is to get as much sand (probability mass) as possible, and Omega's bucket has more sand in it. Monty's unopened door doesn't inherit anything tangible from the opened door, but it does inherit the opened door's probability mass.
-6mhomyack
JoshuaZ110

Think about it this way. Let's say you precommit before we play Monty's game that you won't switch. Then you win 1/3rd of the time, exactly when you picked the correct door first, yes?

Now, suppose you precommit to switching. Under what circumstances will you win? You'll win if you didn't pick the correct door to start with. That means you have a 2/3rd chance of winning since you win whenever your first door wasn't the correct choice.

Your comparison to the roulette wheel doesn't work: The roulette wheel has no memory, but in this case, the car isn't reallocated between the two remaining doors, it was chosen before the process started.

Reply