Lobstrosity

Disclaimer: I saw this posted on another forum and I thought it was worth sharing. I'm not sure how well known this problem is, so if you've all heard it before please feel free to shower me with ridicule and hate me for wasting your time.

Here is the scenario.

There are three closed doors for you to pick from. Behind one of them is a fabulous new car. Behind the other two is a goat. Just to be clear, you don't want a goat.

You pick a door. I then open one of the OTHER doors that has a goat, and say to you "You can keep whats behind the door you selected, plus $100, or switch and take what is behind the other closed door!"

What should you do?

Doubting Didymus

Ooo! I agonised over this piece of crap for five years before I finally worked it out for myself. The answer is: you switch to the other door.

Shall I say why, or leave the magic alive for a bit?

Pierre Bezukhov

Switch. Marilyn Vos Savant (sp?) had an article about that years ago.

Asha'man

I agree, you should switch doors.

I seem to recall studying this question (more than a decade ago), but don't remember the exact explanation for why.

In the initial case, you have a 1/3 chance of being correct.

The question is: do you have a 1/2 chance of being correct for the second decision? Are the odds still 1/3? Or something else entirely?

Doubting Didymus

The odds that you picked the right door can not change after the fact. The chance you have the right door is still 1/3. That means that switching doors is as good as being able to pick BOTH the other doors.

The day I finally cracked it (some arsehole told me the problem, and the answer, but refused to tell me why), I envisioned a different situation: Instead of three doors, there are a thousand. You pick one, and then the other guy opens ALL other doors that are goats, bar only one. That's actually the same situation, but it makes it easier to see that of the doors you didn't pick, the probability that the car is in one of them is 2/3. so, when the guy removes all doors bar the one you picked and the one with the car (thats not a random selection, remember), the probability that the car is behind the other door is the opposite probability form the probability that you picked the right one first: in this case, 2/3.

Thats a really really awful explaination. Sorry, but I'm no maths teacher. (not that many maths teachers can explain maths anyway).

Lobstrosity

This problem is interesting in that if you haven't seen it, it's very easy to convince yourself that the wrong answer is in fact the right one. When I first saw this problem posed, there were quite a few people who just would not accept the true answer no matter how many different ways it was justified. I finally saw it, but it took me a little while (the thing that finally won me was a computer simulation someone did, but I think I would have seen it faster if people had been better at explaining it).

Fine then, since this doesn't seem to be supremely challenging, how about I throw another hand-waving problem into the mix for you to ponder. I doubt any of you will have trouble with it, though, since everyone here is pretty damn intelligent...damn you:

You're sitting at a stop light in your car with a helium balloon floating besides you (you're going to a birthday party or something). When the light turns green you slam on the gas and thusly accelerate forwards. Which way does the balloon drift relative to the car while you're accelerating (if indeed it even drifts at all) and why? You can assume all of the car's windows are rolled up and the balloon isn't touching the roof of the car.

Ensign Steve

When you accellerate, the balloon goes to the front of the car. When you brake, the balloon moves toward the back. The opposite of what you would expect it to do, due to inertia. However, the air around the balloon is heavier than the air inside the balloon, so the air moves behind the balloon when you accellerate, and pushes the balloon forward.

Doubting Didymus

I've actually done the experiment using cups and a pea and someone to shuffle it behind my back. Computers are for the lazy!

I don't know the physics one, but I plan on doing the experiment.

Edit: Slapping myself on the forehead, I realise its just obvious: the air moves to the back of the car, because the helium is lighter. Same would apply if the car was full of water and had a large air bubble in it: the water would slosh to the back, forcing the air to the front. (I won't be doing that experiment.)

theyeti

Actually, that's a pretty good explanation. The important thing to remember is that when Monty opens the door with the goat (it's usualy referred to as the "Monty Hall" problem, after the host of the inane gameshow Let's Make a Deal) he's not doing so randomly. He always picks the one with the goat; he never picks the one with the car if it's there. Because of this, he's giving you information about the likelihood of the other door containing the car, but no information about the door you originally picked.

Alternatively, you can simply work out in your head the combinations of things that can happen, since they're not that many of them. You'll see that two out of three times, switching doors gets you the car. Of course that doesn't explain why, but it should be sufficient to convince a skeptic that it happens.

theyeti

Ensign Steve

Not a true skeptic. That's just anecdotal evidence.