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Goats and cars

There's a prob­lem of­ten used to show the un­in­tu­itive na­ture of prob­a­bil­i­ty, which has be­come very well known.

In that prob­lem a con­tes­tant in a gameshow has to choose be­tween three doors (A,B,C), on one there is a car, on the oth­er two are goat­s.

Af­ter the con­tes­tant choos­es, the host opens an­oth­er door and shows a goat.

Then, the host of­fers the con­tes­tant the chance to switch his closed door for the oth­er closed door.

Should he switch?

The in­tu­itive an­swer is "it does­n't mat­ter", be­cause there's two doors and one car, so it's a 50-50 chance.

But the re­al an­swer is that it does mat­ter, be­cause it's a 33-67 chance!

While it's sim­ple to show this to be the case to a sta­tis­ti­cal­ly-e­d­u­cat­ed dude, it's some­what hard­er for a lay­man.

In fac­t, I think most ex­pla­na­tions suck.

Here's my shot at it:

If you were of­fered the chance to switch be­tween your closed door and the oth­er two closed doors, would you take it?

The in­tu­itive an­swer to that is of course, yes, be­cause it's 67-33 for the car to be on the oth­er two doors.

Now, re­gard­less of where the car is, can the host open one of those two doors and show a goat? Of course, yes.

So, would you feel your odds went down be­cause the host showed one of your two closed doors had a goat be­hind it? No, be­cause he could al­ways do that, and you know there was (at least) one goat there!

So, what dif­fer­ence does it make if one door is open or not?

I don't ex­pect this to con­vince any­one, re­al­ly, but just in case, I have a python im­ple­men­ta­tion of this prob­lem (goat­ :-) if any­one wants it, if em­piri­cism can con­vince you ;-)

taj / 2006-04-03 06:23:

You missed out the important part - if you are in this situation, what should you do? Your options are:

1. stick with the original closed door that you selected

2. change your choice to the remaining (non-chosen, non-opened door).

Answer: you should pick option 2, and change your choice to the remaining door. I think I recall reading about this on another blog a while back, can't remember where.

(PS. how about a comment preview please?)

Roberto Alsina / 2006-04-03 06:23:

Erm... that's all I am writing about ;-)

Quoting myself: "Should he switch?"

Roberto Alsina / 2006-04-03 06:24:

Comment preview... I didn't see that. Maybe I need a comment preview ;-)

It's really up to the pycs guys. I will file a wishlist...

DeanG / 2006-04-03 06:25:

I've heard it referred to as the "Montey Hall Problem."

DeanG / 2006-04-03 06:25:

Dangit.. Monty Hall. Good luck in searching for another Python implementation for that one...seems the OTHER show owns that googleshare.

Roberto Alsina / 2006-04-03 06:26:

Maybe I am weak in the gameshow department, but what other show?

illtiz / 2006-04-03 06:40:

I found this explanation convincing at the time:

After your first pick, you're stuck w/ a goat in 2 out of 3 cases. That's a lot. But in those cases, a switch will win you the car.

nf0 / 2006-04-03 06:51:

nutta lol

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