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 Sep 12 awarded Notable Question May 23 awarded Nice Question Mar 22 awarded Popular Question Dec 27 accepted In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? Dec 27 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? (continued) I assumed this because to interpret "one of my children is a girl" as meaning "exactly one of my children is a girl" would destroy the problem/puzzle in that the answer would be too obviously zero. Also, because I had first seen the question in a book about randomness and probability, I was already in math English mode. Dec 27 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? (continued) I'd award you the 100 bounty points for the clarity of your explanation of the mathematics, if you hadn't added your idea that "Probability puzzles like the one you're asking about rely on these differences of English meaning, rather than on any logical or mathematical problem. In that sense, they aren't really puzzles, they're just tricks." In my case I wasn't tricked -- I immediately assumed that "one of my children is a girl" meant that "at least one of my children is a girl". Dec 27 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? I now, at last, agree that the correct answer is 1/3. (See my question "2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes?" at goo.gl/yyOlK and my comment beginning with "@all: It suddenly hit me that I didn't need that truthful observer.", about half way down the page, under Willie Wong's answer.) Dec 27 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? for or against anyone's answer, but I may have accidentally pressed the left button of my mouse when the cursor was over an arrow. But if you make even a tiny edit to your answer I'll be able to give you an up vote. Dec 27 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? @Rahul: Please read my long comment (posted some 3 hours before your last, beginning with "@all: It suddenly hit me that I didn't need that truthful observer." and tell me what you think. I put it under Willie Wong's answer because I wanted to credit him belatedly with the first (and good) answer. I also want to give you an "up" vote for yours (really for sticking with me), but apparently I screwed up at some time before. When I click the up-arrow by your answer I'm told I've already voted and can do nothing unless the answer is edited. I'm sure that I didn't intentionally do any voting at all Dec 26 accepted 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? Dec 26 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? at-least-one-heads outcomes. I would continue until I had tallied 100 at-least-one-heads outcomes. The result, if anyone's curious, was 70 1-heads, 30 2-heads. As for situation A, its equivalence could be tossing the coins together but only tallying the number of heads (1 or 2) of those outcomes where the nearest coin was heads. An outcome where the nearest coin was tails would be ignored. But of course this is equivalent to just tossing one coin and considering the probability of heads (1/2). Dec 26 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? I saw this equivalence, I realized that an outcome of 2 heads was twice as likely as an outcome of 1 heads. But just so I could report on it, I decided to run a test. I got out 2 Roosevelt dimes (a ten-cent U.S. coin with FDR's head on one side), and a smallish, topless, cylindrical bottle, about 1.5 inches in diameter and 2 inches high, with which I could shake the dimes before dumping them out on a flat surface. The plan was to keep tossing (more like dumping dice from a dice cup) the pair of dimes, ignoring 2-tails outcomes, and tallying the number of heads (1 or 2) in the Dec 26 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? @all: It suddenly hit me that I didn't need that truthful observer. I could more clearly think about the problematic situation B if I 1) Removed he truthful observer (and also Willie Wong's helpful 3rd party with the piece of cloth). 2) Did the coin tossing myself -- tossing both coins myself, ignoring the 2 tails 0 heads outcomes until I had an outcome of at least one heads. This is exactly equivalent to the situation B of my original question (before my unfortunate edits). It may be difficult for some of you patient and not-so-patient people to believe, but as soon as Dec 26 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? (continuing) The toss in A and the toss in B have already been made that produce at least one heads each. THESE are the tosses being reported on. It matters not what the sample space of tossing 2 coins is. The observer is to ignore any toss that does not produce at least one heads. The relevant sample space is the sample space of tossing one coin. Dec 26 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? @Nahul: Sorry. As for your "Well, no, in my last comment..." comment, my answer is "Yes, I disagree". As for your "As for the rest of your comments..." comment, you say, "It makes no sense to ask of the probability of a single one-off event in a vacuum. For example, what is the probability that I have a brother?" I agree. It also makes no sense to ask the probability that one of the coins is heads if I already know that there is at least one heads. That's the given. I know how to compute the probability of tossing 2 coins and getting at least one heads. But that's not relevant to my question. Dec 26 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? (continuing) Both the probability of the toss resulting in A and probability of the toss resulting in B are not under consideration, and are irrelevant to the question I asked. In addition, my position is that both A and B are equivalent to a situation C, in which one coin is tossed, and I consider the probability of the outcome being heads (1/2). You and others don't seem willing to accept either A or B as givens. I think I'll stop here to see how you respond. Dec 26 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? @Rahul: I believe you want to deal with my original question, in its pristine state before I confused things (so said TonyK) by editing it. So lets do that. Also, when we refer to a comment by one of us, lets definitively identify which comment. What I've tried to make clear is that both A and B BEGIN with the coins already having been tossed, resulting in at least one heads. In addition, the observer has already reported as specified -- telling me that at least one of the coins is heads. Dec 25 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? However, I meet one of their children and wonder about the other one. Similarly with the coins. The situations are given, as I've stated. There IS one heads (this is a given), no matter whether the observer looks at only one (it may be misleading to call this one coin "the first coin") or both. In both A and B he supplies me with information which together with some logical deduction on my part, informs me that there is at least one heads. The observer has no strategy. He simply reports. It doesn't matter whether he looked at both coins or only one before reporting. Your turn. Thanks, Rahul. Dec 25 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? whereas I was presenting situations where to begin with there already is one girl. So I visit a family who have two kids. One comes in the living room to meet me. She's a girl. So I'm wondering to myself if the other child is a girl -- and thinking that the probability of the other child being a girl is 1/2. There is no boy-boy. Only girl or boy. Some have set the situation up as examining a random sample of 2-child families. But there is no study. This is a one-off. I just happened to visit this family one day. Examining the gender or number of their kids was not on my mind. Dec 25 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? @Rahul Sure I want what you have to say. In my last comment both of us were dealing only with situation A. You aren't disagreeing with my argument there, are you? As for the observer's strategy, I don't believe I've left room for him to have one. So please clarify. As for probability spaces, I was familiar with them before I asked the question -- I just don't agree that they are of much use here, or have been used in a way that doesn't reflect the situations I've posed. For example, in the one-or-two girls problem, some were allowing for boy-boy to be an original possibility --