I'm baffled by the determination of some of you to overcomplicate this simple problem.
It has already been explained a number of times in this thread.
If I toss a coin I have a 1 in 2, or 50% chance of getting heads. It doesn't matter what I call the coin, it doesn't matter if my previous toss was heads or tails, It doesn't matter if I look at the problem from the point of view of a different coin.
All the fluff has no impact on the straight 50/50 chance of the unknown child being a boy or a girl. It's that easy, everything else is irrelevant.
Even the posts that get the right answer have often got there by the most roundabout ways.
But the point is that it's not as simple as it first appears!Originally Posted by Croydon Bob
If you answer the question using information about the mother (i.e. she has 2 children at least one of which is a girl) then the probability of her having a second girl is 33%
If you answer the question using information about the girl (i.e. she's one of a pair of children) then the probability of her having a sister is 50%
It's that choosing between rows and cells in the table difference I posted about earlier.
The standard solution to this puzzle involves focusing on the mother but when the name Emma-Louise is introduced we focus on the girl. This is why we get different answers hence the apparent paradox. But the two solutions are to slightly different questions - the answer is different depending upon the perspective you take.
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It's not simple coin tossing - there's decision-making going on as well, and depending on the rules of the decision making, there can be different outcomes.
The second question in the 'paradox' is effectively asking "What are the odds that a randomly-picked pair of children are both female given that you have been told one is a girl called Emma"
As an example, if someone was following the rules:
a) Randomly pick a mother of two children who aren't both male
b) Give the name of a female child (pick a name at random if there are two girls)
Then the chances of me being told "This woman has two children and one of them is called Emma" are not the same as if someone was following the rules:
a) Randomly pick a mother of two children who aren't both male
b) Give the name of a female child (if there are two girls and one is called Emma, give her name)
It seems that some people are asking the question
"What are the odds that a given mother of two has a child named Emma?"
when the question seems to be
"What are the odds that a given mother of two has a child named Emma which I will end up being told the name of?"
Without some bias towards declaring Emmas (as in the second algorithm) , I'll only find out that an Emma exists in half the girl/girl pairs, whereas I'll find out about all the Emmas in girl/boy pairs.
Therefore if there are equal numbers of Emmas in boy/girl and girl/girl pairs, the odds are 2:1 that if I end up being told about an Emma, she will have come from a boy/girl pair.
Alternatively put, though an Emma in the population at large is as likely to have a sister or brother, given the constraints of the question it seems I will have a reduced chance of hearing about the existence of Emmas who have sisters. Therefore of the Emmas I hear about, they will be more likely to have a brother than a sister.
As for all the birth order, tallest child, etc, it's basically along the same lines.
If I ask a mother of two unknown children if their eldest child is a girl and they say yes, that's quite different to me being told out of the blue "This woman has two children and the eldest is a girl".
If I make the decision to ask about the eldest child (or there is some pre-existing social convention that the eldest child is always talked about first), that's different from me being given information by someone following a set of rules which I am not party to, (and which in the case of the supposed paradox seem to be focussed on talking first about girls and then an attribute of girls which any girl must posess).
Part 2 still seems to be a question about the mother.The second part is: ‘Jane has two children. One is a daughter, Emma-Louise. What’s the probability that she has two daughters?’
No. Almost everything you (and John) have said is either wrong, or, most often, completely irrelevant.
The question can be rephrased as:
Bob tosses two coins, one is heads, Emma-Louise. What's the probability that he tosses two heads?
The mother is not relevant. The names are not relevant. The sex of the first child is not relevant. It is a straight heads or tails on the second toss. I could see this the first time I read the original post. I'm baffled that others cannot see it.
In your terms, a rephrasing of the question would be
Bob has tossed two coins
We're informed by someone that they're not both tails
What are the chances that they're both heads?
We're then informed that Bob decided on a whim to give one coin he can see heads on a pet name, which we are told
What are the chances now that they're both heads?
1/2.
Once again, nothing you've said changes the facts, it is all just fluff.
Who said anything about the other toss being the second toss.
Perhaps that's were you've slipped up in your thinking.
All I know is that the answer to the first question is indeed 1 in 3, and the answer to the second question is indeed 1 in 2.
Since we agree about the answer to the second question I won't labour the incompleteness in your working.
That "fluff" as you called it is otherwise known as rigour. However you don't need to understand statistics to know that you were wrong. All you have to do is test it.
Two coins, two tallys.
Toss both coins, if you have at least one head add to the first tally. If you have both heads, add to both tallys.
Repeat
I say that soon enough you'll see that the second tally is approaching third of the first. If you're tight you'll see the second tally approaching half of the first.
Here's my results, I use a computer for repetitive tasks so there's a lot of them but I've done the manual process with 20 trials and got a clear result so try that if you don't trust your own programming skills.
Spoiler
I haven't read the whole thread, so apologies if this link has already been posted, but there's a wiki entry on it.
Also, I happened to be reading an article on probability and noticed this problem was on the same site, if you're a bit of a masochist it's here in mind-numbing detail.
Nope. It makes no difference whether it takes place first or second. The result of one coin toss doesn't influence the result of another coin toss whether it took place before or after. Knowing the result of one coin toss does not influence the outcome of another coin toss.
You can confuse yourself with what you call "rigour" all you like but it has no relevance.
I'm not sure it's possible to be certain about the answer to the second question without knowing the details of the mechanism by which one is told the name of a girl, or at least without stating one's assumptions about the likely mechanism.
Specifically, in the case of a girl/girl pair, is the mechanism purely random as to which name is given, or is it biased towards particular names?
As a parallel, if I am told that a woman has two children, and that one of those children chosen at random is female, that seems to be different to just being told that a woman has two children, and that [at least] one of the children is female.
In the latter case, I have to make assumptions about why I am being told that one child is female - the simplest of which is that the agent is following orders to tell me if [at least] one child is female.
In the former case, (and pretending no correlation between gender and height) being told the child has been chosen at random is equivalent to me only being told a woman has two children and my asking an arbitrary question such as "Is the older/younger/taller child female?"
In the latter case, if I am told a child is female and that they are older/younger/taller, that extra information is meaningless unless I know the mechanism by which I end up being told it - if the orders are simply 'choose a pair of children at least one of which is female, pick a female child and then say whether they are the oldest or youngest', my being told their birth order adds nothing at all to my knowledge about the gender of the other child.
There seem to be parallels with the Monty Hall puzzle - there's an agent involved, and knowing that agent's rules of engagement makes a difference as to the correct decisions regarding probable outcomes.
If in the Monty Hall puzzle we had no knowledge of the agent's rules, merely that we chose one door and that they had opened another and there was nothing behind it, we wouldn't be in the same position as knowing they were only allowed to open empty doors - we could easily have been in a situation where their rules were to randomly open one of the two doors I hadn't chosen, and we had just happened to end up in a situation where they had opened one with nothing behind.
Matt, you yourself showed that there are equal numbers of boy/girl and girl/girl pairs which contain an Emily (that is, any Emily in the population as a whole is as likely to have a brother as a sister).
If the rules of engagement of the agent are such that I will always be informed about the Emily in a boy/Emily pair, but I will only have a 50% chance of being informed about an Emily in an Emily/other girl pair, then given that I have been informed there's an Emily there, then that Emily is twice as likely to have a brother as a sister.
The very simplest rules the agent seems likely to be following are:
a) Choose a woman with two children
b) If one child is female, name her, if both are female, name one of them
Those rules necessarily underreport by 50% any girl's name from a two-girl pair compared to girl's names from boy-girl pairs.
In the first we say the "at least one of the coins" is heads. So we're examining the result of both coin tosses. The result of a question regarding the pair of coin tosses certainly has the potential to effect the probability of the the individual. It's called conditional probability.
If I roll a die what is the probability that it comes up 1? 1 in 6 right?
But if I've rolled a die and all I tell you about the result is that it was an Odd number you can calculate the conditional probability that it came up 1 given that it came up as an odd number.
Divide the probability that any die is rolled with both a 1 and an odd result (1/6) by the probability that any die is rolled with an odd result (1/2)
We get one in three.
That's conditional probability. But hey, I don't expect you to follow what just seems like fluff to you.
I do however expect you to respect the empirical approach.
Do the experiment I outlined. Tell me what results you get.
I'm not sure that it does matter that Monty knows which door to open. We do know the door when it was opened didn't show the prize. That's all we need for the conditional probability calculation. Whether Monty fixed it or we got that result by chance and discard the other possibilities makes no difference. After the fact we still know that the probability that the door Monty opened didn't contain the prize is 1.
But yes Tolman, the way you've given the infomraiton does matter.
If the girl you've been told about could be either of the children we get the first question.
BG, GB and GG will all result in you being told that there's a girl.
If on the other hand a child is specified first and you're told that she's a girl then we get the second question.
Under that scenario you could have had one of each but still be told boy.
There's a certain amibiguity in the way the quesitons were initially phrased and I suspect that this is where much of the confusion lies.
Generally, if someone told me that a person had two children, at least one of which was a girl, that would tend to suggest that the person telling me had some undisclosed reason to talk about girls.
Any subsequent extra information given unprompted about a specific girl child in the pair of children doesn't necessarily change my view about the probable gender of the other child - I could be told the age, birth order, hair/eye colour,whether they had a higher or lower IQ than their sibling, but if those are all just titbits of information volunteered unprompted, they don't seem to have any extra value.
On the other hand, if I asked "Is the eldest child a girl?", "Is the brightest child a girl?", and the same for tallest/blondest/sportiest/noisiest, etc, then with each positive reply I could increase my expectation that both would be girls.
In the case of information offered unprompted by someone who had already told me that there was at least one girl in the pair, they could be under instructions to simply choose the girl from a boy/girl pair, or randomly choose a girl from a girl/girl pair and then give me all kinds of biographical information about that specific girl.
In the case of me asking questions, I'm not asking about a specific girl, but about the child which happens [randomly] to fit in a particular position within the pair.
As for the name, assuming conventional naming and ignoring gender-ambiguous names, on being told that at least one child was a girl, I could predict with 100% confidence that at least one child would have a female name, and therefore that if I asked to be given a female name of one of the children in a pair, it could always be done.
Given that it could always be done if at least one child is female, and that it could never be done if both children were male seems to suggest that it having been done tells me nothing more than I already knew.
What the agent in the second question seems to me to be saying is:
"This mother has two children.
At least one is a girl.
I will give you more information about a female child in the pair [selected either by default (in the case of a boy:girl pair) or by some undisclosed mechanism in the case of both being girls].
That child's name is Emily"
The information content of an individual girl's name seem to amount to nothing more than 'This child is female', which I already knew would be the case.
If the agent had begun by saying:
"Hi! I'm called Emily. Don't you think that's just the most beautiful name?"
and had then said:
"Here's a woman who has two children. One of them is a girl called Emily!"
I think the odds on the girl Emily having a sister probably would be 1 in 2, because I'd have a reasonable expectation that my chances of hearing about an Emily in a pair would be equal whether or not the pair was boy/girl or girl/girl.
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