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 Dec25 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 -- Dec25 comment Series that converge to $\pi$ quickly J.M.: Which is the faster, Ramanujan's or the Chudnovsky brothers'? Dec25 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? @Rahul: In A the observer only knows about the one coin he looks at. He tells me that at least one of the coins is heads, which in A must mean that the coin he looked at is heads. IOW "At least one of the coins is heads" DOES imply "The first coin is heads". No? Dec24 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Jonas I don't agree that I left any ambiguity between "one particular child", "at least one child", and "exactly one child". Look at the context: "I had heard that he had 2 kids and one was a girl. I was going to visit him soon and was wondering about the other child. Girl? or boy?" Clearly this implies that I believed he had at least one girl, and had no knowledge about the other child. Dec24 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? @Rahul Thank you for not giving up on me. I thought I'd written the question (before the edits) to indicate that in A and B the coins had already been tossed, with at least one heads in both cases, as reported by the observer. In A, the situation is such that the one coin the observer looks at IS heads; in B at least one of the 2 coins he looks at IS heads. Those are givens. Their probabilities are irrelevant, IMO. Therefore my claim that I receive exactly the same information from the observer in both cases. Dec24 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? @TonyK So what are the probabilities for A and B as of my first edit? I'll now edit again and have the observer look at both coins in B and pick one, about which he will tell me, if it's heads, "There is at least one heads", or if tails, "There is at least one tails". What is the probability of the other outcome being heads or tails, respectively? Dec24 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Jonas "How did you hear that one child is a girl?" I wrote "I had heard that he had 2 kids and one was a girl." I think that implies that someone told me that (or possibly wrote that to me in a letter or email), and that's all I know about my nephew's children. Isn't that the standard interpretation? Dec24 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? With my edit I believe I've removed any possible analogy with the Monty Hall problem. How about it? Dec24 comment 2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? @Willie It seems absurd to me that the probabilities could be different due to the different knowledge the observer has of the outcomes: knowing about only one of the coins versus knowing about both the coins. What he tells me is identical (and truthful) in both situations: "At least one of the coins is a head". Dec23 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 believe that last word should be "girls"? Dec21 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Hendrik Well, I thank you for trying. It seems to me that a test could be run that would show who's right. Say 1000 guesses having knowledge A and 1000 guesses having knowledge B. Dec21 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Hendrik Yes, I look an only one of the coins, but I don't know which one is the one showing Heads. The knowledge I have is exactly the same as the knowledge gained from someone who has looked at both coins but told me only that one is showing Heads. Dec21 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Hendrik But if the coins were identical except for the year they were minted (say 1991 and 1995), I wouldn't be able to tell which one was the one I could see from say, 6 feet away. The only thing I could know is that there is at least one Heads. Rather than a different experiment, it seems the same to me as the one you formulated. Dec21 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Jonas It isn't at all clear that "The original question intends you to imagine a randomly chosen 2 child family from among all families with at least 1 daughter." Instead I imagined I hadn't seen my nephew in a decade, and we hadn't kept in touch. But I had heard that he had 2 kids and one was a girl. I was going to visit him soon and was wondering about the other child. Girl? or boy? I figured the chances of a girl was 1 in 2, and I still do. I found the Wikipedia article linked to by Rawling enlightening. Dec21 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Hendrik Let's say he flips 2 coins at the same time, and they land on the floor. I can see only one of them, and it's a head. So, given that it's a head, the probability of the other coin showing a head is 1/3? Surely not. Now, in my visit to the family, I know nothing about their kids other than there are 2. Further let's assume that there is no reason to think that the first one I see will be a girl rather than a boy. But it is a girl. Given that that kid is a girl, the probability of the 2nd kid I see being a girl is 1/3? Dec21 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Shai Let's say I visit a friend who flips a coin. The first flip is Heads. Are you saying that the probability of the second flip also being Heads is 1/3? Of course not. But then what is the difference between these two coin flips and the successive appearance of that family's two children? Dec21 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Shai But why? Why is (b) different from (a) and (c)? Dec21 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'm hoping that someone will show me why my 3 other examples are different from the case in question. a) the bowl of marbles b) my visit to this family c) my first child is a girl. The second? Dec21 comment In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? @Rawling Say I visit this family. I know they have 2 kids. One of them, a girl, comes into the room. The probability that the 2nd kid is also a girl is 1/2, no? Or let's say my wife and I have our first child, a girl. The probability that the next child we have will also be a girl is 1/2, no? Nov30 comment Is Knopp's “Theory and Application of Infinite Series” out of date? Thanks. I thought that surely there would be developments in this area in the 60 years or more since publication that the modern reader should know about. None? Only a few? Actually, I hope there aren't, because I can see the book becoming one of my favorites.