Timeline for very simple conditional probability question [closed]
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2010 at 16:27 | comment | added | Tom LaGatta | JBL, I think you're right about why we're getting two different answers. At this point, I've seen enough arguments to think your approach is the right one: the opening child is selected independently of the configuration. Here's another one using Bayes' theorem. $P(X=BB | O=B) = P(O=B | X=BB) * P(X=BB) / P(O=B) = 1 * 1/4 / 1/2 = 1/2$. | |
| Aug 21, 2010 at 15:05 | comment | added | JBL | Okay, I finally figured it out. Tom LaGatta (and maybe also Yemon) believe that "a boy opens the door" is meant strictly to rule out the possibility of GG in a Monty Hall-esque way, i.e., without changing the relative probabilities of BG, GB and BB. KalEl, Aaron, Dominic and I all believe that "a boy opens the door" means that of the two children, one is chosen with equal probability to open the door, and it happens that this one is male. The question is ambiguous as to which of these two is intended. | |
| Aug 19, 2010 at 9:18 | comment | added | Aaron Meyerowitz | A town has 100 houses each with 2 kids. 25 BB, 50 BG, 25 GG. You knock on each door and a random child answers. You'd get 50 boys answering, 25 from a BB house and 25 from a BG hose. So the chance of (non answering child=B | answering child was a B)=1/2. | |
| Aug 19, 2010 at 4:08 | comment | added | JBL | Yemon: As far as I understand it, you and Tom LaGatta claim that Pr(X = BB | B opens door) is equal to 1/3 (not 2/3 as I erroneously wrote), from which it appears to follow (from my computation above, corrected) that 1/6th of all two-child families are BB. So, working on the assumption that you haven't made any error, could you explain what I am misunderstanding or doing wrong? | |
| Aug 19, 2010 at 1:14 | comment | added | Yemon Choi | Also, is there a reason why no one's mentioned Monty Hall here? | |
| Aug 19, 2010 at 1:13 | comment | added | Yemon Choi | @JBL: PR(X=BB | B opens door) is not 2/3, because if we take as our state space (gender of child 1, gender of child 2; gender of child who opens door) then there are six possible outcomes but they are not all equally likely. Indeed, (B,B;B) and (G,G;G) each occur with probability 1/4; each of the other 4 configurations occurs with probability 1/8. | |
| Aug 18, 2010 at 22:54 | comment | added | JBL | Tom LaGatta: what is wrong with the following reasoning? Let X be a randomly selected two-child family. Then Pr(X = BB) = Pr(X = BB|B opens door) * Pr(B opens door) + Pr(X = BB | G opens door) * Pr(G opens door) = 2/3 * 1/2 = 1/3. | |
| Aug 18, 2010 at 22:02 | history | closed |
Qiaochu Yuan Robin Chapman Andy Putman Yemon Choi Pete L. Clark |
off topic | |
| Aug 18, 2010 at 19:51 | vote | accept | Not Bayes | ||
| Aug 18, 2010 at 19:45 | comment | added | Not Bayes | okay, done. closing it here would be fine with me. | |
| Aug 18, 2010 at 19:42 | comment | added | Tom LaGatta |
The interviewer's logic is correct. You are simply trying to compute the probability $\mathbb P( one child is a girl | one child is a boy)$. The probability of the event $\{one child is a boy\}$ is irrelevant here to the computation of the conditional probability-- it happened, so use that information! In the sample space formalism ("list all possibilities"), this is given by counting the number of boy-and-girl outcomes and dividing that by the number of outcomes with a boy, so 2/3 is the correct answer.
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| Aug 18, 2010 at 19:37 | answer | added | KalEl | timeline score: 1 | |
| Aug 18, 2010 at 19:37 | comment | added | Nate Eldredge | No, you just have to repost. | |
| Aug 18, 2010 at 19:29 | answer | added | Dominic | timeline score: 0 | |
| Aug 18, 2010 at 19:26 | comment | added | Not Bayes | oh, sorry, I didn't know of that site - that looks like a more appropriate site. Is there an easy way to move it over there? | |
| Aug 18, 2010 at 19:21 | comment | added | Jon Bannon | This question would probably be fine at math.stackexchange.com | |
| Aug 18, 2010 at 19:15 | history | asked | Not Bayes | CC BY-SA 2.5 |