| Gale
Greenlee |
We ask the truth telling Mother of two children: (a) is at least one of your children a girl. She answers, "Yes". or (b) we ask her to tell us the sex of one of her children and she answers, "female". What is the probability the woman is the mother of two girls. (a) , (b) ?? Gale |
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| Dan
Goodman |
This is a famous problem. Like the Monty Hall problem, most people (including mathematicians) get it wrong. I would say there's a fifty fifty chance that the following is the wrong answer, but it's the one I'm going for: The probability of her having two girls is 1/4, the probability of her having two boys is 1/4, the probability of her having one boy and one girl is 1/2 (this is before we know the answer to either question). This is the assumption of the question, although I suspect that these probabilities are factually inaccurate (there are slightly more girls than boys in most societies I think). Both answers to (a) and (b) will be the same, because the information we get from both answers is that she has at least one girl. This rules out the possibility of two boys. So, P(two girls given at least one girl) = P(two girls) / P(at least one girl) = (1/4)/(3/4) = 1/3. |
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| Michael Doré |
Dan - I think the point about the second one is that the mother is asked about a _specific_ child. For example the question might say: what is the sex of your first child? In that case, the probability that both children are girls is the probability that the second child is a girl, i.e. 1/2, since the genders of the children are independent. The same would also apply if we asked the mother to pick one of her two children at random and say what his/her sex is. |
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| Dan
Goodman |
Michael, reading the question literally "tell us the sex of one of her children" suggests that she could choose which child to tell us the sex of, but the logic of maths questions suggests that you're right. It would have been clearer if she'd been asked "tell us the sex of your first child". Incidentally, here is another one of this sort of probability questions: you walk into a town and see a number 50 bus, how many bus routes are there in that town? (Assume that if there are n bus routes, you are equally likely to have seen any one of buses 1 to n.) |
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| Gale
Greenlee |
I think this is the classic example of why you can't rely on Bayes Theorem, the formula P (2 girls)/ P (at least one girl) may or may not yield the P (two girls given at least one girl). There are lot's of opinions out there but my opinion is that the answer, "yes" at situation (a) is not a truthful answer. It's my position that one cannot truthfully say, "at least one of two children is a girl" if that person knows the sex of both children. It's either " they are both girls" or "exactly one is a girl". The answer "yes" could only be true if the respondent knew the sex of only one of the two children, and then the "yes" answer leads to the probability of 1/2 for two girls. So the answer to both (a) and (b) is 1/2. GALE |
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| Gale
Greenlee |
Concerning the bus question. Aren't there > or = to 50 routes? GALE |
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| Gale
Greenlee |
Think about it this way. Consider an elderly woman in a nursing home. She is suffering badly from dementia. She can remember that she has two children and that she has a daughter named Mary. She doesn't remember if Mary is the older or younger child. She can't remember the sex of the other child. The probability this woman is the mother of two girls is 1/2. |
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| Michael Doré |
Gale - "yes" is a perfectly truthful answer. The question simply asked whether one of the two children are girls. There is no obligation to tell the whole truth - only to answer the question. Also your last message is not a fair rephrasing of a). Even though the mother suffering from dementia cannot remember whether Mary is the younger or elder child, the point is that she has a _specific_ child in mind. However in the way you originally phrased a) the mother could report "yes" without having a specific girl in mind. Dan - I agree there certainly is ambiguity in the question. I personally would interpret it as the following: the mother is asked to choose one of her two children at random (with equal probability) and say what his/her sex is. In this case given that she has reported "female" then the answer to the question is certainly 1/2. However I agree that if (say) the mother is the sort of person who will report "girl" if she possibly can and "boy" only if she absolutely has to, then the answer is 1/3. But if the mother is not sexist then the answer is 1/2. Anyway the point is, I think, that we need to make modelling assumptions about the process the mother uses to choose which child to report the sex of. |
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| Gale
Greenlee |
I quote "By Michael Dore on Tuesday, March 11, 2003 - 04:25 pm: Gale - "yes" is a perfectly truthful answer. The question simply asked whether one of the two children are girls. There is no obligation to tell the whole truth - only to answer the question" ____________________________________________ I disagree. There is an obligation for a truthteller to tell the whole truth. Otherwise the events are not mutually exclusive. GALE |
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| Dan
Goodman |
I don't think that we need to make modelling assumptions about the process the mother uses, because if we don't make any modelling assumptions and just use the actual fact that she has reported then we get the 1/3 answer. Gale, re busses: but what is your best guess at the number of bus routes? |
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| Gale
Greenlee |
Dan: I find the bus question interesting. My best guess is 50. But I'm hard pressed to answer why I think 50 is more likely than 51. GALE |
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| Gale
Greenlee |
Michael: You say, " However in the way you originally phrased a) the mother could report "yes" without having a specific girl in mind." If she had no specific girl in mind it would be because she had two girls and then the probability for two girls is 1. Gale |
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| Michael Doré |
Dan - OK look at it another way. We ask the mother to report the sex of one of her children. Now it is true that the fact the mother answers "female" implies that at least one of her children is a girl. However the converse isn't true. That at least one of the mother's children is a girl doesn't guarentee that she will answer "female" to the question. (She might have had a boy and a girl and answered "male".) So the fact she _does_ answer "female" is telling us _more_ than the fact she has one girl. (Although it doesn't tell us anything more with certainty.) It tells us not only that she has at least one girl, but _also_ that she reported "female" despite the fact there was still a possibility (as far as we are concerned, knowing only that she has > =1 girl) that she could have reported "male". This pushes up the chance that she has two girls. So the chance should be more than 1/3. Here is the calculation. When we ask the mother to report the sex of one of her children we assume she picks one at random and reports their sex. So P(she answers female) = P(picks first born).P(first born is female) + P(picks second born).P(second born is female) = 1/4 + 1/4 = 1/2. And P(she answers female and actually has two girls) = P(she actually has two girls) = 1/4. Therefore P(she actually has two girls|she answers female) = (1/4)/(1/2) = 1/2. Of course if we make a different assumption about how the mother chooses which child to report the sex of then that will chance the answer. If she is determined to report "girl" unless she can't then the answer will be 1/3. If she is determined to report "boy" unless she can't then the answer is 1. Gale - what events are not mutually exclusive? The question is a simple yes/no question. An answer yes means the mother has at least one girl. An answer no means the mother has no girls. These events are certainly mutually exclusive. All the mother has to do is answer the question - she doesn't have to say everything she knows, otherwise there wouldn't be any point to the problem. Also regarding your last message - yes I agree but how is that a refutation of my point? |
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| Dan
Goodman |
"When we ask the mother to report the sex of one of her children we assume she picks one at random and reports their sex." I quite agree that the probability is a 1/2 if we assume this, but the question doesn't say she does and intuitively it's an unlikely assumption. Gale, 50 is a good guess. One reason for thinking that it should be 50 is that 50 is the "maximum likelihood estimator" (MLE). That is, of all the choices of n, n=50 makes your observation of a number 50 bus the most likely. The chance that you see a number 50 bus given that there are n routes and n> =50 is 1/n (it is 0 if n < 50). Since 1/50 < 1/n for n> 50, n=50 is the MLE. It's a nice puzzle because people usually say n=100. |
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| Jeremy
Reizenstein |
99 would be an unbiased estimate of n: If X is the number of the bus you see, then E(2X-1)=2E(X)-1=2(1+n)/2-1=n The MLE is not necessarily a good guess. Using this 2X-1 will make you correct on average. |
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| Gale
Greenlee |
A yes answer comes about because of a situation that has a 3/4 probability of occuring. This having occured, one child of each sex has a 2/3 probability of being the situation. The product of these two (3/4) x (2/3)= 1/2 is the probability both happened (one of each sex) (fourth principle). And 1 minus this P is the probability for two girls, which is 1/2. The mother knows the sex of both her children. And so the only truthful answer to the question "is at least one of your children a girl" is, no. Thats how it refuts the last point. |
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| Gale
Greenlee |
That 2x-1 is very interesting. Thanks. GALE |
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| Gale
Greenlee |
I said "The mother knows the sex of both her children. And so the only truthful answer to the question "is at least one of your children a girl" is, no." I don't really mean that. The answer "no", would not be truthfull. But neither would the answer "yes" be the whole truth. Gale |
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Michael Doré
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Dan - I'm not trying to be awkward here, but I really don't see what other assumption is more natural, and (assuming sexism doesn't play a role) I don't see how you can get an answer of 1/3. How would you interpret "we ask her to tell us the sex of one of her children"? I see two possibilities: "please pick one of your children at random and report his/her sex" or (alternatively) "please say 'boy' if you have a boy, 'girl' if you have a girl, unless you have both in which case just pick boy/girl at random (with probability 1/2 each)". Both of these views lead to an answer of 1/2 - I don't really see any other sensible interpretation apart from these two. Gale - if I asked you whether you have at least one hand would you be dishonest in saying "yes" because you know you actually have two hands? Or suppose you were applying for a job and on the application form it said "do you have at least 5 GCSEs" then is the only honest answer no because in fact you know you have 10 GCSEs? No, of course not. The question isn't asking for every scrap of information you know, it's only asking for partial information. |
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| Gale
Greenlee |
Michael: We are on the same side here. I think 1/2 is the only good answer. If you are willing to use partial information when full information is available then the answer 1/3 can be supported. But that is a special case and only interesting in an academic sense. GALE |
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| Michael Doré |
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