# Monty Hall problem



## robert@fm (Dec 14, 2016)

This is a well-known one, see if you can solve it without resorting to Google. 

On a game show hosted by Monty Hall, the grand finale has the one remaining contestant confronted with three doors. Behind one is the grand prize of a new car; behind the other two are goats.

The contestant is asked to pick a door, then Monty (who knows where the car is) opens one of the other two doors to reveal a goat. He then asks the contestant whether he wants to stay with the original door, or swap.

Should the contestant stay or swap, or doesn't it matter?

(Think carefully; even professional mathematicians sometimes get this one wrong.)


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## robert@fm (Dec 15, 2016)

I'm surprised nobody's tackled this one. All it needs it a bit of advanced calculus and tensor mechanics — er, no, I'm kidding. All it needs is a bit of simple probability; any schoolkid could probably figure it. (Hint: you start with a 1-in-3 chance of having picked the right door; what happens after that?)


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## Robin (Dec 15, 2016)

I hate probability questions! Common sense says to me that it shouldn't matter whether you'd already picked a door or not, if one of the doors is taken out of the equation. then it should be a 50:50 chance of the two remaining ones. Car, or not car. But probability rules probably say otherwise!
Unless it's a Schrodingers car, which could be behind either door and is only fixed when you've looked!


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## JazzieJamJay (Dec 15, 2016)

I seem to remember a similar, or possibly exact same question being asked in the movie '21'

He should swap...He now has a 66% chance of winning by swapping to the other door. I don't know how or why....its just what the movies tell me!


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## robert@fm (Dec 15, 2016)

Most people, including Ph.D's, argue that revealing the goat means that there are now only two possibilities as to where the car is, so it doesn't matter if you switch or not, because the probability of each is 50%. The fallacy of this argument is that the two possibilities are not evenly distributed.

In fact there are three equally likely possibilities: contestant picks car, Monty reveals a goat (doesn't matter which), staying wins; contestant picks goat 1, Monty reveals goat 2, switching wins; or contestant picks goat 2, Monty reveals goat 1, switching wins.There is thus a 2/3 chance (which is 67%, not 66%) of winning if you switch. 

If you think that's bad, try Bertrand's Paradox: "If a randomly-chosen chord is drawn across a circle, what is the probability that it is longer than the side of an equilateral triangle inscribed in the circle?". Depending on what you mean by "randomly-chosen", the answer could be 1/2, 1/3 or 1/4.


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## Ditto (Dec 15, 2016)

Is this like a Schroedinger's Cat thing? My brain hurts.


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## HOBIE (Dec 17, 2016)

Goat-Car-Car-Goat    Doors


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## mikeyB (Dec 17, 2016)

Try something easier. 

Two trains, both travelling at 50mph, are catastrophically heading towards each other on the same line. When the trains are 1 mile apart, a super fit fly takes off from the front of one train, lands on the front of the opposite train, and instantly takes off again to do the opposite, and does this without a break until the inevitable crash.

How far has the fly flown before it gets squished?


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## robert@fm (Jan 15, 2017)




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## trophywench (Jan 15, 2017)

I'd need to know how fast a superfit fly flies Robert.

Anyway, he's still dead in the finish which has got to be good - one less pesky fly in the world !

However - I do care about the train drivers.  Did they survive?


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## robert@fm (Jan 22, 2017)

mikeyB said:


> Try something easier.
> 
> Two trains, both travelling at 50mph, are catastrophically heading towards each other on the same line. When the trains are 1 mile apart, a super fit fly takes off from the front of one train, lands on the front of the opposite train, and instantly takes off again to do the opposite, and does this without a break until the inevitable crash.
> 
> How far has the fly flown before it gets squished?


As @trophywench has said, you have omitted one important piece of information; _how fast was the fly going?_ If it went at infinite speed, instantly covering the entire distance between the trains, then it flew an infinite distance; if on the other hand it just teleported back and forth, it flew zero distance. I suspect the intended answer is somewhere between these two extremes.


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