Originally posted by Palynka It's a subtle point, but what you say is not the same as the stated problem. It assumes that the friend would always say that there's at least one H when the coins are different.

If you don't assume that, then conditioning on "at least 1T" is not the same as conditioning on "the friend tells you that there's at least 1T"!

Please see my second post on " is not a good way to explain what he means to say. Let's forget about that for now.

To be perfectly fair, you are entirely correct. I was trying to cut to what I saw as the real fundamental problem with wolfgang's reasoning, which I think can better be fleshed out by assuming that we are conditioning on fact X -- not conditioning on that some friend claims fact X (which is itself a different fact). On that there is strictly a difference in these two cases, you are entirely correct. There's a big difference: in one case you are conditioning on fact X, in the other case you are conditioning on a different fact, the fact that some friend claims fact X to be the case.

But, do you think wolfgang meant to claim strictly that we are conditioning on the fact that a friend claims some fact; or do you think he meant we are conditioning on the base fact itself (by which he brings up a friend's statement as one way the fact could reliably be brought to light). I think he means something like that latter, in which case I think his reasoning is clearly flawed. If he actually meant the former, then maybe my comments are not completely fair to him, and we should be considering the subtlety more carefully like you did.

Originally posted by LemonJello To be perfectly fair, you are entirely correct. I was trying to cut to what I saw as the real fundamental problem with wolfgang's reasoning, which I think can better be fleshed out by assuming that we are conditioning on fact X -- not conditioning on that some friend claims fact X (which is itself a different fact). On that there is strictly a differenc ompletely fair to him, and we should be considering the subtlety more carefully like you did.

There's a big difference: in one case you are conditioning on fact X, in the other case you are conditioning on a different fact, the fact that some friend claims fact X to be the case.

Er...I don't like the way I stated this. I think you know what I mean, but a better statement is that there's obviously a big difference between conditioning on that P and conditioning on that some observer claims that P.

Likewise, you could ask whether the OP is talking about conditioning on that P or really does mean to strictly talk about just conditioning on that some observer claims that P. You're right there's a big difference, but I thought that the OP is talking about conditioning on that P itself (and that the existence of the friend is just anectdotal to how the conditioning on P might be imported).

Originally posted by Palynka It's a subtle point, but what you say is not the same as the stated problem. It assumes that the friend would always say that there's at least one H when the coins are different.

If you don't assume that, then conditioning on "at least 1T" is not the same as conditioning on "the friend tells you that there's at least 1T"!

Please see my second post on " is not a good way to explain what he means to say. Let's forget about that for now.

One more point: that fact that conditioning on P is different from conditioning on that your friend claims that P is not just a subtlety that regards, for example, your second post on page 1. That they are different is not really something subtle at all: in the two cases, you are (strictly, again) conditioning on completely different information. After all, when you consider that P; and that some friend claims that P; neither strictly even entails the other. To get them on remotely the same footing you have to assume yet more information: that the friend is not mistaken; or is not deceiving you; or is a reliable source; etc.

Originally posted by LemonJello One more point: that fact that conditioning on P is different from conditioning on that your friend claims that P is not just a subtlety that regards, for example, your second post on page 1. That they are different is not really something subtle at all: in the two cases, you are (strictly, again) conditioning on completely different information. After ...[text shortened]... rmation: that the friend is not mistaken; or is not deceiving you; or is a reliable source; etc.

Isn't that what I wrote on my second post on page 1? I'm confused as to what you mean here, but if I understand you correctly then I agree entirely.

Personally, I think considerations about whether he's lying or not (and equivalent musings) is entering the realm of puzzle pedantry so I was taking for granted that he wasn't. But you're correct that's an assumption I make for the sake of the puzzle.

I don't see the comments I made about conditioning as being on the same sphere of puzzle pedantry, as they are key to the 'paradox' that the original poster mentioned. For example, in that 2nd post of mine, differentiation between conditioning on #B>1 or Db (I think) makes it clear where the distinction lies.

Originally posted by Palynka Isn't that what I wrote on my second post on page 1? I'm confused as to what you mean here, but if I understand you correctly then I agree entirely.

Personally, I think considerations about whether he's lying or not (and equivalent musings) is entering the realm of puzzle pedantry so I was taking for granted that he wasn't. But you're correct that's an as ...[text shortened]... between conditioning on #B>1 or Db (I think) makes it clear where the distinction lies.

Hello ,
I must say that I am so glad to se the probability post. I have just finished a class doing the same thing . Very interesting concept. Thank you for different in put on the probability concept. Ctina

As often seems to be the case in these things, it's interpretation that's the problem - not the calculation. The confusion seems to arise from what the statement actually means, and the context in which it applies.

For instance, if we are told "a child chosen at random is female", then it's pretty clear that the other child is 50-50 male/female. If the person is asked "do you have any daughters" and the answer is "yes", then there's a 2/3 chance that they also have a son.

But if you are just told that the person says "I have a daughter", then you have to make all sorts of assumptions about what would prompt them to say that. For instance, you could make an argument that they would never have phrased it like that if they had two daughters, and therefore they must have a son as well.

The "overhearing on the phone" version is another way of saying "a randomly chosen child" - if we assume boys and girls are as likely to be on the phone.

Originally posted by mtthw As often seems to be the case in these things, it's interpretation that's the problem - not the calculation. The confusion seems to arise from what the statement actually means, and the context in which it applies.

For instance, if we are told "a child chosen at random is female", then it's pretty clear that the other child is 50-50 male/female. If the pe ...[text shortened]... as likely to be on the phone.

It's not a hard problem if you're precise ðŸ™‚.

Correct and succinct mtthw!

Overhearing a daughter on the phone reduces the possible two child options to;
BG
GB
GG

the child on the phone (random!) is one of the FOUR possible girls and NOT one of the three possible scenaios.

We thus have a 2/4 chance that its GG and a 2/4 chance its BG or GB

Therefore chances of the other child being male is 50%