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[Reelya]

Scenario: You're on a tacky game-show, you've survived the Wheel Of Death, beaten all the other contestants in the elimination rounds, and now you're up to the final challenge of the night.

The host speaks: "Before you are doors number 1, 2 and 3, behind one of the doors is a car, the other two have goats. Which door do you want?"

You pick one of the doors, then the host opens another door to reveal a goat.

He speaks again: "Your door could have the car, or does the remaining door have the car? You have one chance swap doors right now before we reveal where the car is, it's 50/50."

What do you do? Do you stick with your door or swap doors? Does it even matter?

[kaijyuu]

It doesn't matter, assuming there's actually a car. 50/50 either way.

However, this might be one of those trick games, where none of the doors are actually correct. I'd pick a door, but force the host to open all 3, just so I can be certain there is a car back there and not 3 goats.

[Patchouli]

Spoiler: Relevant Wikpedia! (click to show/hide)

http://en.wikipedia.org/wiki/Monty_Hall_problem

[Frumple]

... my question would be what's the reward (is there one?) for picking a door with the car over the door with a goat.

Because if you're reading that somewhat literally, the host's asking which door you want. Not what's behind it. If that's the case, unless there's something special about the individual doors, any door will do.

[MonkeyHead]

It can be shown using simple probability that you are better off swapping your final choice. Its a classic probability problem.

Darn, Ninja'd

[The Darkling Wolf]

I stick with my door.

I'd rather know that I'd never picked the winning prize than I'd picked it, and then changed my mind about it.

[Akura]

I'd probably keep my choice. If I win a car, I win a car. If I win a goat, huzzah! I won a goat.

[Karlito]

I'd probably want a goat more than a car, honestly.

[MonkeyHead]

You cant milk a car.

[The Darkling Wolf]

Well I was assuming that I wanted the car more, based on the wording of the puzzle.

[kaijyuu]

Spoiler: Relevant Wikpedia! (click to show/hide)

http://en.wikipedia.org/wiki/Monty_Hall_problem

Well this is quite wacky. I see how it works on the math angle, though.

*muses*


The reason it works is the revealed door will never be one you picked. As such, if you pick a goat door, the other goat gets eliminated. If you pick the car door, there's still 1 goat left.

The instances where it is in your favor to switch are when you picked a goat door. If you initially picked the car door, you're still at a 50/50 chance to get it by switching (as far as you can tell). However, if you picked a goat door initially (which you have a 2/3 chance of doing), you will ALWAYS win by switching. Thus, switching is in your favor.

[Reelya]

Well the solution is to swap, assuming you take home the car (a classic major prize in game shows, the goat is a placeholder for losing).

2/3 times your first choice was a goat. Then, the host gets rid of the other goat, meaning that you're guaranteed to get the car by swapping. 1/3 times you DID pick the car, and get a goat by swapping, but overall the odds are better to swap.

The "50/50" line was a red herring, since you picked out of 3 to start with.

This is actually an example of how humans are awful at processing even simple probability questions. This one stumped maths professors at major colleges when it was first proposed and they refused to accept the answer. We have the benefit of wikipedia now though.

Spoiler (click to show/hide)

Many readers refused to believe that switching is beneficial. After the Monty Hall problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming that vos Savant was wrong (Tierney 1991). Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy. According to the French magazine Le Figaro Magazine, Paul Erdős, the most prolifically published mathematician of modern times, found it very hard to believe the result of his own calculations, and admitted he was not fully convinced until a simple computer simulation confirmed the predicted result.[1]

[Lectorog]

I'd stick with my door. I know about

Spoiler: Relevant Wikpedia! (click to show/hide)

http://en.wikipedia.org/wiki/Monty_Hall_problem

but whatever. I've never been a fan of the mathematical branch of probability.

The "50/50" line was a red herring, since you picked out of 3 to start with.

If you make another completely random choice, disregarding all previous information, there is a 50/50 probability.

[Reelya]

You don't have to do that in this case, because picking again at random loses information that was encoded in the actions of the host. The door that the host reveals is modified based on what door you picked, and what the host knows is behind the doors.

The fact that the coded information exists is evident in that you have a 2/3 chance of profiting by swapping. That's why the 50/50 line is a red herring.

[MetalSlimeHunt]

A similar problem that demonstrates human fallibility concerning probability is the Birthday Paradox, which shows that any group of 23 has a 50% chance of containing two individuals with the same birthday and any group of 53 a 99% chance of the same, despite there being 365 days to be born on.