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Brain Teaser

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booper54

Brain Teaser

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A woman contestant on a game show is told that she has to choose one of three doors for her prize. Behind one door is a brand new car. Behind each of the other two doors is a goat. The game show host knows which door the car is behind. The woman chooses a door. Before he opens the door she chose, the game show host opens one of the other two doors and shows that there is a goat behind that door. The remaining closed doors are the one the woman chose and the other unopened door - one having the car and one having the other goat. Before opening a second door, the game show host gives the woman an opportunity to switch her door from the one she chose to the other unopened door. Should she switch? Does it matter?
RickU

Re: Brain Teaser

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I want to say that it doesn't matter. When there were 3 she had a 33.3% chance of choosing the door with the car. (I'm working under the presumtion that she wants the car...she may well have no use for it and wish for the goat)...At 2 doors her chance becomes 50/50 and so choosing to switch wouldn't make any any difference statistically. In Vino Veritas
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Mr. P

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Re: Brain Teaser

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It does not matter. She already beat the 33.3% odds and gained a 50/50 just by the one door having the goat. Like multiple choice tests...it is always best to stick with your first instinct.Mr. P. The one thing of which I am positive is that there is much of which to be negative - Mr. P.I came to get down, I came to get down. So get out ya seat and jump around - House of PainHEY! Is that a ball in your court? - Mr. P
booper54

Re: Brain Teaser

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Both of your answers are what I thought initially; but then I saw this:"There is a 1/3 chance that she picked the door with the car, and a 2/3 chance that one of the other doors has the car. When the guy reveals one of the goat doors, these statistics do not change. So if she keeps the door she is with, she has a 1/3 chance of getting a car where as if she switches doors, she has a 2/3 chance of getting a car.i understand it seems counterintuitive, but since the reveiling of a door does not create a seperate event, the odds from the original event stay true. Look at it this way...lets label the doors 1, 2, and 3. Lets say door 1 has the car and 2 and 3 has goats. The outcomes are as follows:a) she picks door 1, door 2 or 3 are shown to be a goat. If she swtiches, she loses, if she stays, she wins. b) she picks door 2. She is shown door 3 to be a goat. If she swtiches, she wins. If she stays she looes.c)she picks door 3. She is shown door 2 to be a goat. If she switches she wins. If she stays she losesIn two of the events, switching doors will make her a winner. So based on this, the odds of winning if she swtiches is 2/3 where as the odds of winning while staying are only 1/3."Very interesting... any thoughts?
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Mr. P

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Re: Brain Teaser

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But does not the fact that she is allowed to pick again mean that she is picking out of only two choices? Since a door was revealed and a new selection was allowed...the chances are indeed 50/50% because it is a new and separate event.Mr. P. The one thing of which I am positive is that there is much of which to be negative - Mr. P.I came to get down, I came to get down. So get out ya seat and jump around - House of PainHEY! Is that a ball in your court? - Mr. P
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Re: Brain Teaser

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It's a trick! Don't fall for it!! One goat in the hand is worth 2 unopened doors. I think she should choose the door that has been revealed, so she KNOWS that she's getting a goat.
booper54

Re: Brain Teaser

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ScrumfishVery true! mr. pThat's a good point. The flaw in the answer is that there is a separate event after the host opens the first door. Therefore, it's like starting over at that point, and so there's a 50/50 chance.
booper54

Re: Brain Teaser

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Apparently this is a statistics problem.Consider this:"Assume there are 1000 doors. After she selects one at random, the host opens 998 other doors that she didn't pic. Behind each of these doors is a goat. Now, the only two doors that are left are the door she picked and one other. Do you really think she has a 50/50 shot at the car if she merely keeps the door she originally picked? Of course not. The reason is that no matter which of the 1000 doors she picked, he could eliminate 998 doors that were incorrect. She still has only a 1/1000 chance of being correct. As for the seperate event, NOTHING happened. The host merely showed doors that were NOT the correct door. Although there are now only two doors left, that fact does not increase her odds of being correct when she actually made the choice. Make sense?"I had to really grasp that before I believed it. But now that I believe it, it's hard to trust myself to equally consider the other side now.What a mind-bender!
RickU

Re: Brain Teaser

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Now that's interesting. But I think the analysis breaks down when you pose it in that fashion. Her odds of choosing the correct door on the first try are 1/3 in the first example. In the 2nd example the odds are 1/1000 - So your odds of choosing the correct door ON THE FIRST TRY are vastly different. In the 1/1000 example, I would choose the 2nd door (the one I didn't originally choose). I haven't taken statistics, but logically, having 998 choices eliminated changes the landscape dramatically. Having 1 of 3 eliminated in this fashion does not change it drastically enough.I'll ask my sister to weigh in later on this week. (She's got her Masters in math.) In Vino Veritas
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