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The Monty Hall Paradox


The Doctor

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Let's see how many of you can get this...

 

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A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door."

You begin by pointing to door number 1. The host shows you that door number 3 has a goat.

 

Do the player's odds of getting the car increase by switching to door number 2?

 

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I'll refrain from giving the answer for a while, just to see who can get it. Explain your reasoning, too, if you can. Use spoliers, if you don't mind - if someone wants to give it a shot, seeing the result will ruin it.

 

For those that don't know, spoiler tags are these:

 

[spoiler]SPOILER TEXT HERE[/spoiler]

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OK, genius time;):

 

I'm pretty sure there's some evil trick question-style thing here I haven't thought of, but at the same time, I've gotten burned by thinking too deeply about easy questions before, discarding the obvious answer because it was too easy to reach.

 

There are three combinations, right (actually six, or 3x2x1:

 

C G1 G2

C G2 G1

G1 C G2

G2 C G1

G1 G2 C

G2 G1 C

 

...But let's focus on the cars):

 

CGG

GCG

 

We know it's not GGC, because the show host has demonstrated that Door #3 contains a goat. It has to be CGG or GCG.

 

So the chance of success is 50%. If we choose the other door, it's still 50%, right? The odds don't change.

 

So... No, I don't see how your odds would increase by switching doors.

 

 

Damn. Wrong.

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Actually, you have 2/3 chance to pick a car if you switch you choice.

 

You first have 2 out of 3 chances to pick a goat and 1 out of 3 chances to pick a car the first time around. By switching doors, you also switch the odds. Basically if you had a goat (2 out of 3 times), you'll end up with the car.

 

 

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