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Post by invisible on Jun 24, 2008 8:01:26 GMT -7
This is based on a game show which I never watched. There are three closed doors. There is a goat behind each of two doors and a car behind the third. You get whatever is behind the door you pick. You pick a door. The MC says, "I'm going to give you another chance." and opens one of the doors you did not pick. There is a goat behind it. Assuming you are expert in probability theory, what do you do, and why? I'd suggest you consider this problem before looking at the second set.
There is kind of a reverse version of this problem. Suppose a mother tells you, "I have two children, one is a girl." What are the odds on the other child being a girl?
However, suppose a Father tells you, "I have two children, the oldest is a girl." What are the odds the other child is a girl?
These are actually difficult questions. I did not understand them until a geneticist colleague explained the second set to me.
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Post by zrct02 on Jun 24, 2008 21:11:23 GMT -7
I am no expert. However, I think I'd change doors. If a stranger came in at this point, he'd have a 50% chance of getting it right. I only had a 33% chance when I chose my door. It appears to me that if I change doors, I move up to a 50% chance of being correct. This is a WAG.
In the first child case I'd say there is a 50% chance. It seems to me that these are totally independent events. The other child case is significantly different as we appear to be looking at an ordered set. In that case, I'd have to guess at 25%. This is another total WAG.
I'll be very interested to hear the real answers.
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Post by invisible on Jun 25, 2008 7:46:15 GMT -7
The Monty hall problem generally ends with the question, "Should you change doors?" As said, you had 1/3 chance the first time, and now you have a 1/2 chance. I think you should make a new decision, ignoring as best you can your first decision. Maybe flip a coin.
On the second problem, you know that there are two children, one a girl. You do not know the birth sequence. The possible birth sequences are: GG, GB, and BG. So the chance of two girls is 1/3, even though the chance of a girl in any one birth is 1/2.
In the third problem you know the sequence. The girl is the oldest, and the possibilities are reduced to GG and GB, and the chance that the second child is a girl is 1/2.
This is a simple example of the principle that the more you know about a situation, the better your chance of making correct predictions.
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Post by Galvin on Jun 25, 2008 13:39:56 GMT -7
The original chances were one in three. Once he opened the door they were ostensibly one in two.
But were they?
I picked a door and the host opened one I did not pick. Since he knows which one has the car he is certainly not going to open that one right off the bat. He thus opens a different one, i.e., one with a goat behind it.
He then asks me if I wish to make another selection. Since game shows are designed to generate suspense and prolong the agony and because the host has a certain amount of time to fill, it is a given that he would not open the one with the car behind it if it was my first choice.
It is also a given that he will first open a door with a goat behind it but not the one I picked even if I picked wrong. There are two of those doors he can choose from but only one with the car behind it.
There is no guarantee that the first door I chose is the correct door but the probability of its being the correct one may have risen slightly merely because of that fact, i.e. that the correct door will never be opened after the first guess.
Perhaps sticking to my original guess has a slightly higher probablility of riding home in style rather than in a goatcart.
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Post by invisible on Jun 25, 2008 14:12:56 GMT -7
For the Monty Hall game to be fair, the MC cannot know what is behind each of the two remaining doors. The goats and car were placed by someone else, and that someone tells the MC, via his little earphone, which door to open. Just being given another chance, I think, would predispose you to change your original decision.
As a young person, I made my college money by raising goats. So I would just give a come to feed call. The goat behind the door, being frustrated at being unable to come to feed, would start bleating, thus I would know which door it was behind. ;D
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Post by Galvin on Jun 26, 2008 3:53:24 GMT -7
If the game were played in the simple form you describe then it would be a case of a simple 33.333% chance of choosing the correct door.
This would only increase to a 50% chance as soon as the first door was opened, provided it was NOT one that the contestant had asked to be opened and was one of the two with the goats.
Open the one he asked for and a goat strolls out, he immediately loses. Nor can it be the one with the car or else the game is over because the contestant has won on the first round.
Ergo the host MUST know which doors have which items behind them in order to be able to set up the final question.
The way the shows are actually run, the first door opened has to be one that was not picked and also has to be one with a goat behind it because the actual game is based solely on the second guess once the offer of a second guess is given.
The first guess only serves to provide an excuse to open a door that was not chosen in order to set this up.
In fact, if done completely logically and without the chicanery given in the description, the game could easily be won on the first guess and the only door opened the right one. It could also be lost twice as easily on the first guess as well.
But the host must open a wrong door. Why? Obviously in order to prolong the game. If he did not, the contestant would either immediately be presented with whichever goat or car happened to be behind the door of choice and the second guess would never occur.
By stipulating that the game show host open one of the two doors that were not chosen in order to then ask the contestant if he wishes to change his mind, the first choice of doors is made superfluous to the game even if the contestant does not change his choice of doors from that originally made.
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Post by Patch on Jun 26, 2008 23:28:19 GMT -7
That all sounds interesting, but ever since I read the original post, I can't get politicians, media, and extremists off of my mind. to me, someone telling me that they have 2 kids and 1 is a girl, would easily lead me to assume that the other is a boy. That would not necessarily be the case though if someone wanted to be dishonest without lying. I often wonder why the courts say, The truth, the whole truth, and nothing but the truth when they won't let you answer anything outside of the EXACT question. So MR. Smith (who has 2 daughters) do you have a daughter? Yes I do, in fact, I have........Mr Smith! Answer the question!
Other than that, I agree with Galvin.
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Post by trimtab on Jun 27, 2008 9:56:46 GMT -7
......... if someone wanted to be dishonest without lying. I often wonder why the courts say, The truth, the whole truth, and nothing but the truth when they won't let you answer anything outside of the EXACT question. So MR. Smith (who has 2 daughters) do you have a daughter? Yes I do, in fact, I have........Mr Smith! Answer the question! Other than that, I agree with Galvin. Just like the man sitting on a bench next to a dog and a stranger asks "does your dog bite?" The man answers no, the dog suddenly bites the stranger and the man responds with "that's not my dog".
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shaun
New arrival
Posts: 13
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Post by shaun on Jul 8, 2008 22:27:59 GMT -7
I am no expert. However, I think I'd change doors. If a stranger came in at this point, he'd have a 50% chance of getting it right. I only had a 33% chance when I chose my door. It appears to me that if I change doors, I move up to a 50% chance of being correct. This is a WAG. In the first child case I'd say there is a 50% chance. It seems to me that these are totally independent events. The other child case is significantly different as we appear to be looking at an ordered set. In that case, I'd have to guess at 25%. This is another total WAG. I'll be very interested to hear the real answers. Definitely I would also have changed the doors.
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Post by probligo on Jul 20, 2008 16:16:53 GMT -7
Shaun, here was me thinking that "free flight upgrade" had something to do with the model aircraft disciplines I enjoy 1. Game shows are never "fair". 2. Therefore to try and apply normal rules of probability are futile. 3. Other than that - the host would chose a door with a goat behind it? Mostly, but I can imagine the "let me choose for you" rule being applied... Alternatively, "you can not win the prize behind the door I select" could well turn up the car - just occasionally so that viewers can see that there is a car behind the door. 4. Game shows are never "fair". 5. Imagine - in the 3 minutes that he can fill in with exhortations, suspense, and choosing another door, the car is shifted and replaced by a goat. Or was it? There was a real hooha in Aussie about 5 years back when a major prize was lost on an "incorrect" answer. It went to Court, and regrettably the game show won because one of the conditions of participation was that the decision on the night is final. So you can be right, the host can say "Wrong" and you have no recourse...
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Post by invisible on Jul 21, 2008 7:19:15 GMT -7
The assumption of the problem is that the show is fair. If not, the problem is not interesting. Not talking reality here.
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shaun
New arrival
Posts: 13
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Post by shaun on Jul 22, 2008 3:38:30 GMT -7
We should talk reality either that's a better option
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Post by invisible on Sept 25, 2008 13:27:05 GMT -7
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