Skeetin
Talk pages on articles
editSkeetin;
Please do not pose questions in the article about the article... Use the talk page, which is linked at the top. I have moved your question to the talk page and have answered it there. Charles 02:16, 14 July 2006 (UTC)
Boy or Girl Paradox
editSkeetin, your points on the "Boy or Girl Paradox" are simply incorrect. I'm not saying that you can't try to apply a different interpretation to the word "boy"; but you have to ignore the accepted definition of the word "boy" to do so. The interpretation that "boy" refers to one of the children in the family is, indeed, required by that accepted definition. As literally stated, all of the questions in that article refer only to the gender of children. So the possibilities you brought up do not help to understand it.
But there are possible ambiguities, that cannot be removed by definition. And some depend on how the problem is worded. I brought the discussion here, because the discussion page for the article is not really for this kind of discussion.
We are dealing with these three variations of the problem, all asking for the probability that a random two-child family includes a girl as one of the children:
- Q1: You are told the older child is a boy.
- Q2: You are told that the family was selected from all two-child families that include a (child who is a) boy.
- Q3: You are told that the family includes a (child who is a) boy.
The answer to Q1 is clearly 1/2. The answer to Q2 is clearly 2/3. Q3 is ambiguous at best. It is ambiguous if one insists he knows the intent of the question, beyond what the words say (unlike what you did, where you chose to blindly apply a non-standard definition of a word). Such a person is misinterpreting the event that he is trying to enumerate. This is clear, because both Q1 and Q2 are possible interpretations of Q3.
The event in Q3 is not that the family was chosen from all two-child families that include a boy. It is that some incomplete determination of the family's makeup was made, and as a result "includes a boy" was reported. Since it was incomplete, it is possible for "includes a boy" to be true, but not be what is reported. You are enumerating these cases, and you should not.
That the reporter intended the observation to mean Q2 is not a required interpretation, like it was when we take "boy" to mean "one of the children." The answer can only be determined by either (1) Assuming how that determination was made, or (2) creating a sample space based in an unbiased way on the methods that could have been used to make that incomplete determination. Assuming Q3 is the same as Q2 is an assumption, no more or less valid a priori than assuming it is Q1. So the assumption method leads to ambiguity. By creating the unbiased sample space, the answer you get is 1/2 (more below).
You are also wrong about the case of having seen just one of the two children. In Q1, you don't have to have used age to order the children. Any unbiased (w.r.t. gender) ordering of the two children will suffice. That includes "the child you saw first," if you can somehow eliminate the case where the two were seen simultaneously. If you saw just one, simultaneity is not an issue. Having seen only one child only in Q3 reduces it to Q1, replacing "older" with "seen first." OR, you could employ the larger sample space as you tried to describe it; but you have to do actually do it (as opposed to describing it and stating a result, as you did) and do it correctly. These are the equiprobable sets of outcomes in that sample space:
| Older child | Younger child | Child Seen |
|---|---|---|
| (B) Girl | (B) Boy | (B) Younger |
| (B) Boy | (B) Girl | (B) Older |
| Boy | Boy | Older |
| Boy | Boy | Younger |
Those crossed out are the ones we must eliminate. Those that remain are the ones we must keep. There are four equiprobable outcomes remaining, and in two of them the family also includes a girl. That makes the odds 1/2. (And feel free to change the third column to "not seen." That will move the cross-out rows from those marked (A) to those marked (B), with the same result.)
In the case where you have to assess the ways the determination could have been made, there are only two unique determinations possible that tell you information about gender and number of children, but don't completely describe that information in the family: "includes a boy" and "includes a girl." Here is that sample space if those are the possible observations, including the probability of occurrence since they are no longer equiprobable:
| Older child | Younger child | Information | Probability |
|---|---|---|---|
| Girl | Boy | includes a boy | 12.5% |
| Boy | Girl | includes a boy | 12.5% |
| Boy | Boy | includes a boy | 25% |
You now have to use Bayes Theorem, and get that the probability is (12.5%+12.5%)/(12.5%+12.5%+25%) = 1/2. And BTW, this corresponds to the similar solution to Bertrand's Box Paradox, where you determined that a box held a gold coin, rather than that a family included a boy.
Finally, if you want references that describe the 1/2 answer, see Bar-Hillel & Falk, Section 4.3 in Grinstead and Snell's probability course, or a similar problem published in The American Statistician.
Response to your Reposne on my talk page
editSkeetin, you may not be aware that Wikipedia articles can change frequently. When I wrote the section "I disagree with the solution," its question 2 was worded differently: "A two-child family has at least one boy. What is the probability that it has a girl?" That is the Q3 that I described for you. You also might note that I didn't say 2/3 was wrong, just that it was ambiguous and that there is a better answer. And I didn't say that the solution to Bertrand's Box Paradox is 1/2; that answer applies to the version of the Boy or Girl Paradox that is set up similarly to it. It has four boxes, not three. And that I didn't say that the answer to all three questions is 1/2. The answer to my Q2 is 2/3.
It is clear that you are not going to accept a reasoned solution, no matter how correct it is. You only want to accept what some poeple (not all, as my references show) who cannot look past Q2 in establishing a solution think is the only approach. Many experts see it the way I do, and it is trivial to point out where those who don't are overlooking a critical point. That is what I tried to do for you.JeffJor (talk) 19:19, 3 February 2009 (UTC)
It should be trivial for you to revise the article to include your alternate interpretation, citing one of the "many experts" as a reliable source. If those who don't interpret it the way you do have overlooked the truth, then you should find it very easy to correct the article immediately. One can only wonder why you have not done so already, if it is so trivially easy to prove the article incorrect. Since we're dealing with singular numeric quantities, if 2/3 is not incorrect, then it is correct. If there is a better answer than 2/3, then 2/3 is not correct. Either 2/3 is correct, or some other quantity is correct. Your revision will show this, and the world awaits it with baited breath. Skeetin (talk) 04:48, 7 February 2009 (UTC)