The Monty Hall problem
Sep 12th 2013 | 2:59 am
#1
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From: Mallorca
The Monty Hall problem
Things are not always as they seem.
The Monty Hall problem
Imagine a TV game show not unlike Deal or No Deal in which you choose one of three closed doors and win whatever is behind it.
One door conceals a Cadillac - behind the other two doors are goats. The game show host, Monty Hall (of Let's Make a Deal fame), knows where the Cadillac is and opens one of the doors that you did not choose. You are duly greeted by a goat, and then offered the chance to switch your choice to the other remaining door.
Most people will think that with two choices remaining and one Cadillac, the chances are 50-50.
The most eloquent reasoning I could find is from Emerson Kamarose of San Jose, California (from the Chicago Reader's Straight Dope column in 1991): "As any fool can plainly see, when the game-show host opens a door you did not pick and then gives you a chance to change your pick, he is starting a new game. It makes no difference whether you stay or switch, the odds are 50-50."
But the inconvenient truth here is that it's not 50-50 - in fact, switching doubles your chances of winning. Why?
One door conceals a Cadillac - behind the other two doors are goats. The game show host, Monty Hall (of Let's Make a Deal fame), knows where the Cadillac is and opens one of the doors that you did not choose. You are duly greeted by a goat, and then offered the chance to switch your choice to the other remaining door.
Most people will think that with two choices remaining and one Cadillac, the chances are 50-50.
The most eloquent reasoning I could find is from Emerson Kamarose of San Jose, California (from the Chicago Reader's Straight Dope column in 1991): "As any fool can plainly see, when the game-show host opens a door you did not pick and then gives you a chance to change your pick, he is starting a new game. It makes no difference whether you stay or switch, the odds are 50-50."
But the inconvenient truth here is that it's not 50-50 - in fact, switching doubles your chances of winning. Why?
Sep 12th 2013 | 3:41 am
#2
FreeThinker
Joined: Nov 2009
Posts: 9,702
From: the moors and levels
Re: The Monty Hall problem
It's easy to see why but somewhat more difficult to explain.
Essentially, if your door conceals a Goat then Monty has shown you exactly which door conceals the Cadillac.
It is twice as likely that your door conceals a goat as a Cadillac - so you should change your choice.
Essentially, if your door conceals a Goat then Monty has shown you exactly which door conceals the Cadillac.
It is twice as likely that your door conceals a goat as a Cadillac - so you should change your choice.
Sep 12th 2013 | 4:51 am
#3
Thread Starter
Banned
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From: Mallorca
Re: The Monty Hall problem
It's easy to see why but somewhat more difficult to explain.
Essentially, if your door conceals a Goat then Monty has shown you exactly which door conceals the Cadillac.
It is twice as likely that your door conceals a goat as a Cadillac - so you should change your choice.
Essentially, if your door conceals a Goat then Monty has shown you exactly which door conceals the Cadillac.
It is twice as likely that your door conceals a goat as a Cadillac - so you should change your choice.
Very good. Impressed!
Sep 12th 2013 | 5:29 am
#4
FreeThinker
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Posts: 9,702
From: the moors and levels
Re: The Monty Hall problem
I'm a bit of a nerd.
Sep 13th 2013 | 12:20 am
#5
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From: Mallorca
Re: The Monty Hall problem
Well here it is:
Sep 13th 2013 | 6:14 am
#6
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Joined: Jun 2009
Posts: 1,217
From: Valencia area
Re: The Monty Hall problem
It's easy to see why but somewhat more difficult to explain.
Essentially, if your door conceals a Goat then Monty has shown you exactly which door conceals the Cadillac.
It is twice as likely that your door conceals a goat as a Cadillac - so you should change your choice.
Essentially, if your door conceals a Goat then Monty has shown you exactly which door conceals the Cadillac.
It is twice as likely that your door conceals a goat as a Cadillac - so you should change your choice.
Sep 13th 2013 | 7:36 am
#7
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Joined: Mar 2011
Posts: 1,518
Re: The Monty Hall problem
It's easy to see why but somewhat more difficult to explain.
Essentially, if your door conceals a Goat then Monty has shown you exactly which door conceals the Cadillac.
It is twice as likely that your door conceals a goat as a Cadillac - so you should change your choice.
Essentially, if your door conceals a Goat then Monty has shown you exactly which door conceals the Cadillac.
It is twice as likely that your door conceals a goat as a Cadillac - so you should change your choice.
Sep 13th 2013 | 8:11 am
#9
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From: On Top of the World
Sep 13th 2013 | 9:52 am
#11
On the road again.
Joined: Aug 2008
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From: On Top of the World
Re: The Monty Hall problem
To think I used to be a Maths whizz kid at one time,....but now...
Sep 13th 2013 | 5:54 pm
#12
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Joined: Sep 2005
Posts: 3,984
Re: The Monty Hall problem
This is one of the best puzzles I've seen, got me head scratching for a good while THANKS Ami for the OP. i couldnt get my head around the previous explanations sadly for me, proving I am not a whizz!
Fantastic !
Here' how I figured it (finally).. Having had to split out the 2 goat option:
Choice. ...... Odds. .....Change. .....Dont change
Car............... 1/3 ............. Lose. .............. Win
Goat. ............ 1/3. ............. Win. ............. Lose
Goat. ............ 1/3. ............. Win. ............. Lose
Total. ......... 100%. ........ 2 wins. .............1 win
Its absolutely amazing that you can double, yes double your chances to win just by changing... Its so counter intuitive.... An amazing puzzle. I should have done the break down straight away rather than scratch a groove in my head
Made my day thanks again
Jon
Last edited by Jon-Bxl; Sep 13th 2013 at 6:02 pm.
Sep 13th 2013 | 7:53 pm
#13
FreeThinker
Joined: Nov 2009
Posts: 9,702
From: the moors and levels
Re: The Monty Hall problem
The thing to remember is that your original choice is twice as likely to be a goat than a car and this fact does not change throughout the game.
When the other goat has been eliminated, you are left with a choice of two, your original choice and one other.
Remembering that your original choice is still twice as likely to be a goat than a car then it follows that the only other possible choice is twice as likely to be a car than a goat.
When the other goat has been eliminated, you are left with a choice of two, your original choice and one other.
Remembering that your original choice is still twice as likely to be a goat than a car then it follows that the only other possible choice is twice as likely to be a car than a goat.
Sep 13th 2013 | 8:19 pm
#14
On the road again.
Joined: Aug 2008
Posts: 17,507
From: On Top of the World
Re: The Monty Hall problem
Still not convinced myself.
I keep comparing it to the old coin puzzle where a coin is tossed ten times and comes down as heads the first nine times.
That doesn't alter the fact that it's still an even money 50/50 bet on the last throw, or at least that's what I've always been given to understand.
I keep comparing it to the old coin puzzle where a coin is tossed ten times and comes down as heads the first nine times.
That doesn't alter the fact that it's still an even money 50/50 bet on the last throw, or at least that's what I've always been given to understand.
Sep 13th 2013 | 8:24 pm
#15
Joined on April fools day
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From: 30 miles from a decent grocery store.
Re: The Monty Hall problem
I call shenanigans.. 1 in 3 chance change or not.
