quote:Something's definitely wrong with this reasoning. There's some sneaky mathematical slight of hand going on somewhere. I'm working on figuring out what it is now.
But the even more clever person would say it is 13/27. That's because out of all the possibilities of two child families when the day of the week they were born is included happens to be 196 (2 sexes * 7 days * 2 sexes * 7 days). Of those, 27 have at least one boy born on a Tuesday. And 13 of those have two boys. Mathematically, that's sound. (Also, he would build up an immunity to iocane and put the poison in both goblets)
quote:It seems to me that the slight of hand goes like this. You're taking numbers from the general population, and then applying them to a situation that has already been narrowed down to a subset of the general population. The numbers 13, 27, and 196 have no bearing whatsoever on the gender of this guy's other kid, because we've already specified that the guy has one boy.
Originally posted by JoshuaD:
quote:Something's definitely wrong with this reasoning. There's some sneaky mathematical slight of hand going on somewhere. I'm working on figuring out what it is now.
But the even more clever person would say it is 13/27. That's because out of all the possibilities of two child families when the day of the week they were born is included happens to be 196 (2 sexes * 7 days * 2 sexes * 7 days). Of those, 27 have at least one boy born on a Tuesday. And 13 of those have two boys. Mathematically, that's sound. (Also, he would build up an immunity to iocane and put the poison in both goblets)
code:#DEFINE BOY = 1
#DEFINE GIRL = 2
$BoyGirl = 0;
$BoyBoy = 0;
for 1..100000 {
children[0] = Random() %2;
children[1] = Random() %2;
if ( children[0] == BOY && children[1] == BOY) {
$boyBoy++;
} elsif (children[0] == BOY && children[1] == GIRL) {
$boyGirl++;
} elsif (children[0] == GIRL && children[1] == BOY) {
$boyGirl++;
} elsif (children[0] == GIRL && children[1] == GIRL) {
#do nothing
}
}
print $BoyBoy / ($BoyGirl + $BoyBoy)
quote:This is not an equivalent statement. It is not known whether it was the first or the second coin (in your example) that came up heads on Tuesday (not that Tuesday matters anyway)
An equivalent question would be - I flipped two coins. The first one was on Tuesday and came up heads.
quote:In other words, what's the probability of one special case (two boys) being true after eliminating another special case (two girls)? And because there's not so many possibilities to begin with, those special cases really skew the possibilities A LOT, which is why you turn out with the probability being a third instead of a half.
I have two children, and they're not both girls. What's the chance that they're two boys?
quote:Now it might be a bit clearer why the special case is important. Your GIVEN is actually introducing the day of the week that one of the children was born on as an additional filtering factor. The few families in which both children are born on a Tuesday are counted only once in your filtered set of results, even though there are actually two boys in that family who meet your GIVEN, and that's why the probability of the other child being a boy is just a bit less than 50%.
You have access to a database of all families in America. You do a filter to get all the families with two children. Now, you ask yourself: what proportion of those families have two boys, GIVEN that the family has one boy who was born on a Tuesday?
quote:The summary of those possibilities still leads you astray. Because the problem doesn't explain the *reason* that the guy mentions Tuesday, or why he mentions one of the children is a boy -- and this is very very relevant to determing how it affects the probability.
Originally posted by Jordan:
I'm sick of blanching pistachios for tonight, so I took some time out to put a summary of all the possibilities into a spreadsheet.
quote:Unfortunately, the initial formulation is just ambiguous enough to be misread, which is why so many people go for the 50:50 option by mistake when the actual answer is one-in-three.
Given that I have two children, and at least one of them is a boy, what is the probability that I have two boys?
quote:If the day of the week really were irrelevant information, this would essentially reduce to the first question and the probability of the other child being a boy would be a third. In actuality, you're now restricting the sample to those in which there are boys who were born on Tuesday, and if you do that you get a result which is much closer to a half simply because there are now more possibilities, and more of them in which there are two boys.
Given that I have two children, and that at least one of them is a boy who was born on a Tuesday, what is the probability that I have two boys?
quote:I'm going to wait till tomorrow to see if you're referring of an á la Monty Hall "the host knows something" situation, some sort of unavoidable initial conditions, or simply assuming that the man might be trying to trick us somehow by randomly selecting the information he wants to tell us. (You should know that I tend to approach problems like this assuming point children and perfectly spherical quiz-show hosts, so the latter possibility just isn't going to cross my mind.
[T]his isn't a pure math problem -- it involves psychological reasoning about the motivation of the guy to give you this information.
quote:Yes. But that's not the question. The question, rather, is: I tossed a coin twice, and it came up heads at least once. What are the chances that both tosses came up heads?
If I tell you I tossed a coin twice and it came up heads the first time. Wouldn't you agree that there's a 50/50 chance that the second toss was also heads?
quote:The point is that you don't know whether the result you have comes from the first or second toss.
In a single run of two coin tosses, how can knowledge of the result of the first toss possibly affect the probability of the second toss?
quote:I don't know what "no psychological factors" means: does it mean the guy randomly selects a kid's gender to reveal?
But I think it's pretty safe to say that the "puzzle" here fairly means us to assume both an equal chance of boy/girl births and no psychological factors.
quote:This selective reporting can be done with coin-flips if you don't want to cloud the issue with gender politics. I can easily construct a program simulating coin-flips and using any of the 3 logics I listed, I could completely confuse your attempts to figure out the probability.
Aris, what you are missing is that the fact of the sexes of the children is independent of any choice being made by the person making the statement. Unless you are positing that some misogynist fathers would exclude themselves by being unable not to say they have two boys when they do, but in that case, you would be better off factoring in female birth rates and twin factors into the equation.
quote:Um....As far as I can tell, you're confusing this with an actual Monty Hall problem.
If he randomly selected a kid whose gender he wanted to reveal, then the fact he mentioned it was a boy, leaves the other one's chance at 50%, not 33%.
quote:No. It means that we are invited to assume the guy's motivations and methods are irrelevant. He has two kids, one of whom is a boy born on Tuesday. Full stop.
I don't know what "no psychological factors" means: does it mean the guy randomly selects a kid's gender to reveal?
quote:No, I think you confuse it for such. You believe the information he gives affects the chances of the remaining options, like the showhost removing one door, increasing the possibility of the remaining door containing the prize.
As far as I can tell, you're confusing this with an actual Monty Hall problem.
quote:Yes. Assuming he was being random with his choice of kid whose birth/gender information he revealed, the odds he has two boys are 50%. All you guys saying it's 33% are wrong. My math coincides with common sense in this: No matter how many further info he gives, (hey, it's a Capricorn redhaired boy born on Tuesday, on a moonless night, while the wolves howled and the omens looked favourably down upon him) it won't budge this from 50%.
But in this problem, John Doe walks up to you and says, "I have two kids, one of whom is a boy born on Tuesday," and you are then asked to compute the odds that he has two boys.
quote:Okay. 50% then. The kids were born before he gave *any* information about them, so ANY information he gives about the gender, day of birth, zodiac signs whatever, won't budge this percentage from 50%.
"He has two kids, one of whom is a boy born on Tuesday. Full stop."
quote:We're looking at combinations, not permutations. There are only three combinations of two children: GG, BG, BB. We've eliminated one. There are only two combinations left.
A clever man would then say it is actually 1/3, since there are 4 possibilities of a two child family: GG, GB, BG, and BB. Since GG is eliminated, there's only 1/3 chance
quote:Tom, you're starting from the assumption that the day of the week is irrelevant; instead, take it as another piece of information.
TomDavidson:
Why is it mathematically sound that 13 of the 27 families with a boy born on Tuesday will have two boys? As 1/3 of all two-child families with one boy have two boys, there are thus 9 families in that result set.
9/27 = 1/3.
In other words, the specific day of the week is irrelevant.
quote:However, as it was actually expressed, you're left wondering if the guy decided to tell us about a particular child before asking us, instead of intending to tell us if he does (or doesn't) have at least one boy. In that case, the probability is 1/2.
There exists a family with exactly two children, at least one of whom is a boy. What is the probability that both of the children are boys?
quote:As I said last night, I suspect more people would arrive at the "correct" answer if the question were phrased right.
I have two children. The particular child I'm thinking of right now is a boy. What is the probability that the other one is a boy?
quote:Here is the complete list of possibilities:
Originally posted by TomDavidson:
Again; why is it the case that of the 27 families with two children who've had a boy on a Tuesday, 13 have two boys? That's simply not true.
quote:Just checking - you realize this was not part of the original question, right?
And the most powerful piece of information yet, that the eldest child is a boy, eliminates a huge selection of combinations from consideration and increases the probability to 1/2
quote:Sorry, I should have quoted or linked.
Originally posted by DonaldD:
quote:Just checking - you realize this was not part of the original question, right?
And the most powerful piece of information yet, that the eldest child is a boy, eliminates a huge selection of combinations from consideration and increases the probability to 1/2
quote:
Brian:
Would it change the odds if he said 'my oldest child is a boy'?
quote:First, I'm really glad you're looking at the data carefully.
Originally posted by Pyrtolin:
Both boys being born on Tuesday needs to be counted twice, because you don't know whether you're talking about the first child or the second child. So this really brings it back to 14 / 28 or 50%.
quote:Only one absolute case? Sure. But that case is twice as likely to occur because there are two ways that it could be arrived at- one for each boy that the original asker might be referring to.
there's only one case in which this happens.
quote:I'm afraid not. What we're counting are families, not how many ways he could be asking the question.
Pyrtolin:
Only one absolute case? Sure. But that case is twice as likely to occur because there are two ways that it could be arrived at- one for each boy that the original asker might be referring to.
quote:If you go by Aris' interpretation of the question, the conditional (i.e. the whole thing that makes the question mathematically interesting) disappears; every question we've examined so far reduces to, "What's the probability that a child is a boy?" and the answer in every case is ½.
As Aris has noted- what he's talking about matters, not just the basic distribution.
quote:The subtlety in Aris' interpretation is that we aren't told how the man in question came to be telling us that he had one son, so the question may be "fixed" before you even start; the stricter formulation isn't ambiguous and is the more interesting interpretation that I (and others) are going by to get our figures.
A family has two children. Given that at least one of them is a boy (or, given that one of them is a boy who was born on a Tuesday), what is the probability that both of the children are boys?
quote:Oh, sweet! I knew you'd get it soon.
Tom:
Hm. In working this out for myself, it turns out to be quite amusing to substitute "born on Tuesday" with "prefers red to blue." And, indeed, I misunderstood. *laugh* Once you have any distinguishing characteristic beyond sex, this breaks down the same way.
quote:You're quite close here to grasping it intuitively. In the same way that saying that one of the children is both male and the eldest fixes the first column (by definition of "eldest"), saying that one of the children is both male and born on a Tuesday fixes most of the first column, except for the one case where both children were boys born on Tuesday.
Pyrtolin:
What you're saying would be true if he said "My oldest child is a boy" , because that resolves a specific, relevant element the problem that was otherwise unknown, in the same way that if you were just looking at the B/G matrix, saying that the first child is a boy would resolve the probability to 50/50.
quote:Yes.
edgmatt:
We are not asking 'what are the chances that this second child will be born a boy' we are asking the question "what are the chances that the man fathered two boys (since we know that he already fathered one)."
quote:You're dangerously close. If the numbers game is getting in the way, you can get quite close to understanding the why of it intuitively instead of mathematically.
I haven't quite got the "Tuesday" part of it yet, but I think it works from the same principle. Something about subtracting the time where BOTH are boys and BOTH are born on Tuesday figures in, but I'm not that strong on this sort of thing.
quote:No, don't pollute my beautiful maths with tricksy semantics!
JoshuaD:
I suspect the following two cases are distinct. Do you agree?
quote:Hee. Badvok, Google the problem we discuss at the beginning of this scenario, the Monty Hall problem.
The gender of the unknown child is not dependent on the gender of the known child (excepting any biological factors).
quote:Elder Boy
If you have exactly two kids, and one is a boy, then either the elder is a boy, or the younger is a boy. And everyone agrees that in both of those cases, it becomes 50:50 as to the sex of the other.
So why are the actual odds 1:3 with that level of information?
quote:The Monty Hall problem is totally different because there is a direct causal link.
Originally posted by TomDavidson:
quote:Hee. Badvok, Google the problem we discuss at the beginning of this scenario, the Monty Hall problem.
The gender of the unknown child is not dependent on the gender of the known child (excepting any biological factors).![]()
quote:What is the causal link? There is either a goat behind the door or there is not. Opening a door does not "cause" a goat to be placed; it simply removes one possibility from the set of unopened doors.
The Monty Hall problem is totally different because there is a direct causal link.
quote:Nope, it doesn't "cause" the goat to appear but the contestant's choice directly affects Monty's choice - which in turn affects the odds faced by the contestant (edit to clarify: because Monty will always pick a goat).
Originally posted by TomDavidson:
quote:What is the causal link? There is either a goat behind the door or there is not. Opening a door does not "cause" a goat to be placed; it simply removes one possibility from the set of unopened doors.
The Monty Hall problem is totally different because there is a direct causal link.
quote:True, but that isn't the probability space. John has at least one son, so he can either have two sons or a son and a daughter - 50:50. The gender of one child doesn't alter the gender of the other child.
In the same way, telling someone that one of your two children is a boy removes one possibility from the set of possible children (namely, that you have two girls).
quote:What question? John states that he has one son! So the probability space only concerns the other child.
Originally posted by JoshuaD:
Badvok: It does alter the probability space. You are going to be disregarding 1/4 of all fathers of two children as you ask this question.
quote:And how exactly does the goat transform into a car? Higher math?
Originally posted by TomDavidson:
quote:Um....As far as I can tell, you're confusing this with an actual Monty Hall problem.
If he randomly selected a kid whose gender he wanted to reveal, then the fact he mentioned it was a boy, leaves the other one's chance at 50%, not 33%.
In the original Monty Hall problem, it matters whether Monty knows which doors have goats behind them because he is required to open a door that contains a goat. If he were not required to do so, and in fact selected doors at random, then it's true that his opening a door has no effect on the probability that the original door selected contains the car.
quote:Read my long post above.
Originally posted by Badvok:
quote:What question? John states that he has one son! So the probability space only concerns the other child.
Originally posted by JoshuaD:
Badvok: It does alter the probability space. You are going to be disregarding 1/4 of all fathers of two children as you ask this question.
quote:Badvok, you are ignoring that birth 'order' matters. You don't know whether 'John' is talking about his elder child or his younger child. There are 3 possible sequences of birth in this case: first born is Girl, second is Boy; first born is Boy, second is Girl; first born is Boy, second is also Boy.
What question? John states that he has one son! So the probability space only concerns the other child
quote:The answer to that question is simply 50:50. There is no other answer, we are simply asking for the probability a child is a boy or a girl.
John Doe comes up to you and says: "I have two kids. One of them is a boy. He was born on a Tuesday." What are the odds that he has two boys?
quote:Birth order cancels itself out. You are only counting boy-boy once - it should be twice Boy1-Boy2 and Boy2-Boy1.
Originally posted by DonaldD:
Badvok, you are ignoring that birth 'order' matters. You don't know whether 'John' is talking about his elder child or his younger child. There are 3 possible sequences of birth in this case: first born is Girl, second is Boy; first born is Boy, second is Girl; first born is Boy, second is also Boy.[/QB]
quote:Yes, and?
Originally posted by JoshuaD:
The odds of me having two children of the same sex is 1/2: [B G], [G B], [B B], [G G].
quote:I'll throw the same back at you!
Originally posted by JoshuaD:
Please try to refrain from knee-jerk responses.
quote:Eh? I think I listed those.
Originally posted by DonaldD:
You forgot about:
Youngest+Boy:Eldest+Girl
Eldest+Boy:Youngest+Girl
= 6 combinations
quote:Nope, you listed their inverse cases only.
Eh? I think I listed those
quote:To focus on this just a little further - if Boy1 and Boy2 are not temporal distinctions, but rather actual names, think of it like this:
Birth order cancels itself out. You are only counting boy-boy once - it should be twice Boy1-Boy2 and Boy2-Boy1.
quote:That was kinda my point. If the specific boy mentioned is older, then the odds are 50:50. If the specific boy mentioned is younger, then the odds are 50:50. Do I have that right so far?
As you can see, there is one chance in three, because the case of "Elder Boy/Younger Boy" actually collapses the two potential cases "Boy 1 older/Boy 2 younger" and "Boy 2 older/Boy 1 younger." When you know which one is older, you can specifically say "Boy 1 is older" and thus eliminate the special case.
quote:You are very possibly correct, this could become a situation where the teacher becomes the student. I'll get back to you tomorrow after having a proper think.
JoshuaD:
Jordan: I was up all night thinking about this, and I'm now nearly certain that you're partially wrong. :-)
quote:PSRT: Actually, we do. Right now, we are working with the simplified population of father with at least one boy.
Well, DonaldD, we don't really know what the population in question is
quote:Since you don't know whether this kid is the oldest or the youngest, this does not seem to me to narrow the possibilities beyond the generic "has two kids at least one of whom is a boy".
- Joe comes up to you and says: "I have two kids." A boy then walks up and says "Hi Dad" to Joe.
quote:Not from the question as posed to us. If we wish to specify which set of parents our hypothetical father has been drawn from, we can do that, and then get an exact answer. Without specifying, the question as worded is ambiguous enough that there are multiple correct answers, depending upon interpretation.
PSRT: Actually, we do.
quote:Really? What the hell of an answer will people with two sons give you? This question is meaningless and unanswerable for people with two sons.
Originally posted by JoshuaD:
[QB]
Now, same scenario. A room full of 1,000,000 dads who each have two children and at least one is a son. Instead I first ask, because I'm trying to kill time, "What day of the week was your son born on?" and receive an answer.
quote:Other of what? If someone with two sons says "I have at least one son", then asking if your other child is a boy is meaningless question because there's no "other". He wasn't referring to any particular boy.
I then continue to ask "Is your other child a boy?" How many will answer yes?
quote:Again a meaningless question for fathers with two sons.
Scenario 5: I have a room of 1,000,000 fathers who have 2 children where one child is a boy. I stand them in a line and ask them each "What day was your son born on?" If answer any day other than Tuesday, I ask them to leave.
quote:If the father is referring to a specific child, then the odds are 50:50.
the specific boy mentioned is older, then the odds are 50:50. If the specific boy mentioned is younger, then the odds are 50:50.
quote:Because when a father says "atleast one of my sons" is a boy, the possibilities don't collapse to
If the only two choices lead to a 50:50 chance, why would the intermediate odds be different just because we don't know which one it is yet?
quote:Not relevant:
Originally posted by Aris Katsaris:
Look at conditional probability please.
Probability(A given B) = (Probability of A AND B) / Probability (B)
Probability(two boys, given atleast one boy) = Probability(two boys) / probability(atleast one boy) = (1/4) / (3/4) = 33%
quote:In this case the events are not linked and therefore conditional probability does not apply.
When in a random experiment the event B is known to have occurred, the possible outcomes of the experiment are reduced to B, and hence the probability of the occurrence of A is changed from the unconditional probability into the conditional probability given B.
quote:Badvok, you went through all that effort, and you just repeated the semantic issue in the end. By limiting the statement to a very specific child, yes you reach the 1/2 value. But your statement above is exactly equivalent to saying "And if we collapse the problem space by saying my son John is a boy we get". Do you see how this is different from the interpretation "at least one of my children is a boy"?
And if we collapse the problem space by saying Child1 is a boy we get:
quote:The probability of one boy is 1 - this is stated in the question.
Originally posted by Aris Katsaris:
What he's arguing wrongly is that he thinks this means the probability of two boys is independent of the probability of one boy.
quote:But that is not the question! The question is "What is the probability the other child is a boy if one is a boy?".
Originally posted by DonaldD:
Now take that same interpretation, which in your case is exactly equivalent to 'either child1, child2 or both are boys' and apply it to your 8 combinations and see what you get.
quote:This conditional probability calculation requires the events to be independent (ie, random, not linked) so it very much could apply.
In this case the events are not linked and therefore conditional probability does not apply.
quote:Yes, sort of: if "one of them is a boy" not "child1 is a boy".
But that is not the question! The question is "What is the probability the other child is a boy if one is a boy?".
quote:It is "Child1 is a boy". John has declared the gender of one of his children - this child can be labelled as Child1. The question is then: What is the probability the other child (labelled as Child2 so as to mark it as a separate and distinct entity to Child1) is also a boy?
Originally posted by DonaldD:
quote:Yes, sort of: if "one of them is a boy" not "child1 is a boy".
But that is not the question! The question is "What is the probability the other child is a boy if one is a boy?".
quote:Not relevant?? That's what conditional probability is all about. That's pretty much the ONLY relevant thing.
Not relevant:
quote:No, the question is what is the probability that *BOTH* are boys if "at least one" is a boy.
What is the probability the other child is a boy if one is a boy?
quote:That's what Probability of (A given B) means. That you assume the B happens and it. To calculate (A given B) with the formula i gave, you calculate the probability that B would have to happen (WITHOUT you knowing it did happen).
The probability of one boy is 1 - this is stated in the question.
quote:LOL, I give up, I'll let you all head off to kindergarten now while I get on with some real work.
Originally posted by Aris Katsaris:
Math proves you wrong.
quote:If he did pinpoint to a particular child, you could label it.
Originally posted by Badvok:
quote:It is "Child1 is a boy". John has declared the gender of one of his children - this child can be labelled as Child1.
Originally posted by DonaldD:
[qb]quote:Yes, sort of: if "one of them is a boy" not "child1 is a boy".
But that is not the question! The question is "What is the probability the other child is a boy if one is a boy?".
quote:Yep, sorry.
Originally posted by DonaldD:
I think Badvok just got a little frustrated, Aris. Patience.
quote:Nope it doesn't but you have now labelled them 1st and 2nd instead
Again, Badvok, look at the possible combinations from a first born/second born perspective:
1st. 2nd
Girl Girl
Girl Boy
Boy Girl
Boy Boy
That is it. By naming the boys in the last case (labelling them if your prefer) you don't make that case more probable.
quote:Since John Doe basically volunteered the information without being probed, to me this would be the same as the following hypothetical situation (ignoring day of the week for now):
John Doe comes up to you and says: "I have two kids. One of them is a boy. He was born on a Tuesday." What are the odds that he has two boys?
quote:You transformed GG to kG where k is the known BOY.
Originally posted by Badvok:
[GG,GB,BG,BB] to [snip] [kG,kB,kG,kB].
quote:It doesn't matter whether the known is a boy or a girl, the known is simply a known and becomes a constant that no longer affects the probability.
Originally posted by threads:
quote:You transformed GG to kG where k is the known BOY.
Originally posted by Badvok:
[GG,GB,BG,BB] to [snip] [kG,kB,kG,kB].
quote:No, no, NO! A million times NO, you don't ask that!
I have a room full of 1,000,000 dads. Each is the dad of two children. I ask each dad to tell me the gender of one of their children.
quote:I'm getting close to tears here. We've all repeated a hundred times that asking dad to name the gender of one of his children is DIFFERENT to asking him about whether he has atleast one boy.
Based on the logic by Donald, Aris, and others, if the Dad said that he had at least one boy, then there is a 66% chance he has a boy and a girl.
quote:The probability of two boys here is unambiguously 33%.
The only way the problem works out to 2/3 is if we restate the problem as follows:
John Doe: I have two children.
Me: Is at least one a boy?
John Doe: Yes.
In this case the probability of two boys is 66%.
quote:1/2 = the same as whether any individual kid is a boy.
Originally posted by Pete at Home:
Badvok, think of this:
Situation 1: Man A tells you he has 2 kids. What's the possibility that his oldest kid is a boy?
quote:Now that is a totally different question and is more like the Monty Hall problem because you have made a selection (the oldest) that directly affects the probability. You have now narrowed the possibilities to 3 (eByB,eByG,eGyB) and hence the probability that the oldest is a boy is 2/3.
Situation 2: Man B tells you that he has two kids, and that at least one of them is a boy. What's the possibility that the oldest kid is a boy?
quote:I don't blame people that don't get boy Tuesday, he's counter to intuition. The 7x7x2x2=196 possibilities are btw too much to keep in mind, so it gets even more confusing.
But I still don't get boy Tuesday.
quote:Badvok, you still don't know which child is 'known'. The father is a black box. And since you don't 'know' that, the child is not 'known' in fact.
If the known is a boy the probability of both being boys is equal to the probability the unknown is a boy.
quote:I guess I can't see how you can possibly get this from the original problem statement of John Doe walking up and declaring he has a boy.
You don't ask each dad to tell the gender of one of their children. YOU ASK THEM WHETHER THEY HAVE atleast one boy.
quote:Which is pretty much what I argued at the whole first page of this thread, so we're in agreement here.
"The only way I see your logic working is if the problem is restated with probing questions being asked, instead of information declared."
quote:It sounds like we don't disagree. From the problem statement, there is no way I would assume that the father only wants to reveal the existence of a boy. All we can tell from the problem statement is that the father has chosen one of his kids, and revealed that this particular kid was born on a Tuesday and happens to be a boy.
IF the father is randomly selecting the gender of one his kids to reveal, there's 50% chance that he has two boys. But IF the father only wants to reveal the existence of a boy, there's 33% chance that he has two boys.
quote:We know the existence of a boy. And the existence of a boy IS interdependent with the probability that a specific child is a boy.
"We know one is a boy."
quote:The combination "one girl and one boy" is twice more likely to occur in a population than the combination "two boys".
"Therefore there is only one single sole distinct individual solitary unknown gender value and that can have only one of two possible values."
quote:Yes, that's what I argued in the first page of this thread.
It sounds like we don't disagree. From the problem statement, there is no way I would assume that the father only wants to reveal the existence of a boy. All we can tell from the problem statement is that the father has chosen one of his kids, and revealed that this particular kid was born on a Tuesday and happens to be a boy.
quote:Nope, sorry. It is still 1/2. No matter how you re-phrase it. There is still only one unknown that can have one of only two values.
Originally posted by Ciasiab:
Badvok
If the problem were stated as follows:
John Doe: I have two children.
Me: Is at least one a boy?
John Doe: Yes.
Would you agree the probability of two boys changes from 1/2 to 1/3?
quote:Yep. But that is the wrong question.
Originally posted by Aris Katsaris:
The combination "one girl and one boy" is twice more likely to occur in a population than the combination "two boys".
quote:No I think out of 100% dads with two boys 100% will have at least one
Again, man: out of 100% dads with two boys: 75% will have at least one boy. But only 25% of the dads will have two boys.
quote:Well yes, but turning the question around doesn't change the number of unknowns.
Originally posted by DonaldD:
Badvok:
"I have two kids. One of them is a boy".
Do you agree that this statement is effectively the same as:
"I have two kids. They are not both girls"?
quote:Correct. Bear with me.
Well yes, but turning the question around doesn't change the number of unknowns.
quote:Except, that it's really really not.
'The combination "one girl and one boy" is just as like to occur in a population (of fathers with at least one son) as the combination "two boys"'.
quote:Yes.
Originally posted by DonaldD:
I assume you agree that each case has an equal chance of occuring. I also assume you agree that the structure of the puzzle implies that all these births predate the father's statements and as such the father's statements can have no effect on the probability of each combination.
quote:Do you want to rephrase that - I don't think it is asking what you wanted to ask. If he has no daughters then he can have only two sons.
Now, given that you are presented with a father of two who claims that he has no daughters, what are the remaining possible family structures available?
quote:Either two boys or a boy and a girl.
Originally posted by DonaldD:
Now, given that you are presented with a father of two who claims he does not have two daughters, what are the remaining possible family structures available?
quote:
Originally posted by Badvok:
Yep. But that is the wrong question.
The answer to the original question is 'The combination "one girl and one boy" is just as like to occur in a population (of fathers with at least one son) as the combination "two boys"'.
quote:Let's get even simpler here since you missed that the 25% was a subset of the original 75%
quote:No I think out of 100% dads with two boys 100% will have at least one
Again, man: out of 100% dads with two boys: 75% will have at least one boy. But only 25% of the dads will have two boys.![]()
quote:John is clearly giving information about one of his two kids. Regardless of how much information John gives you about this particular kid, the odds of the other one being a boy or a girl is 50-50.
John Doe comes up to you and says: "I have two kids. One of them is a boy. He was born on a Tuesday."
quote:Unless both of his kids were born on a Tuesday. From your perspective you cannot be clear that he's talking about a specific one until he tells you that he's talking about a specific one. Intuition and common sense don't apply here; if the information isn't explicitly given, you can't imagine that you have it.
John is clearly giving information about one of his two kids. Regardless of how much information John gives you about this particular kid, the odds of the other one being a boy or a girl is 50-50.
quote:After some more careful parsing, I think you're right—I was wrong. But I think you may be partially wrong about why I was wrong.
JoshuaD:
Jordan: I was up all night thinking about this, and I'm now nearly certain that you're partially wrong. :-)
quote:Where the heck were you 3 pages ago?!?
Intuition and common sense don't apply here; if the information isn't explicitly given, you can't imagine that you have it.
quote:Basically what I was doing, but addressed to Joshua's hypothetical million fathers instead of just one. It's kind of hard to know who's saying "yes"!
Ciasiab:
How about: "Do you have at least one son who was born on Tuesday?"
quote:Intuition works with quantum mechanics.
When intuition doesn't work (I'm looking at you, quantum mechanics!) then the only result I get is a pounding headache.
quote:No, it does not. You still do not know which of his children is a boy. Saying, "I have two children, at least one of whom is a boy," leaves you with a 33% chance of having two boys.
The second statement does not remove one possibility from the two unknown gender probability space. It alters the probability space to be that of only a single unknown gender.
quote:
"I have two children."
Result set:
Boy/Boy
Girl/Girl
Boy/Girl
Girl/Boy
quote:
"At least one of them is a boy."
Result set:
Boy/Boy
Boy/Girl
Girl/Boy
quote:
"The boy I mentioned was born on a Tuesday."
Result set:
Boy Tuesday/Boy Sunday
Boy Tuesday/Boy Monday
Boy Tuesday/Boy Tuesday *collapsed
Boy Tuesday/Boy Wednesday
Boy Tuesday/Boy Thursday
Boy Tuesday/Boy Friday
Boy Tuesday/Boy Saturday
Boy Tuesday/Girl Sunday
Boy Tuesday/Girl Monday
Boy Tuesday/Girl Tuesday
Boy Tuesday/Girl Wednesday
Boy Tuesday/Girl Thursday
Boy Tuesday/Girl Friday
Boy Tuesday/Girl Saturday
quote:And I have been trying (and obviously failing) to point out that it doesn't matter which one is a boy just that one is a boy and therefore the other is either a boy or a girl. Two options only = 50:50 chance of each.
Originally posted by TomDavidson:
You still do not know which of his children is a boy.
Seriously. I have explained to you several times why this is the case.
quote:Except that you're wrong. Seriously. You really are.
And I have been trying (and obviously failing) to point out that it doesn't matter which one is a boy just that one is a boy and therefore the other is either a boy or a girl.
quote:Your numbers are not wrong in that sense.
Originally posted by Aris Katsaris:
We have given you the numbers, Badvok.
100 fathers with 2 children: With random even distribution 75 of these have atleast one son, but only 25 of them have two boys.
This means that only 1/3 of fathers with atleast one son have two sons.
Can you pinpoint to us which one of these numbers you don't accept?
quote:Right- and given the overall distribution of choices, there are two chances that that unknown child will be a girl and one that it will be a boy, because boy-girl families are twice as common as boy-boy families.
But you are just solving the wrong problem and hence using the wrong numbers for the problem. We have ONE father, ONE son, and ONE other child who's gender we do not know.
quote:That's not a comparable situation. When you put two kids into the room, the possibility existed that you put two girls in as well. It's not till _after_ you randomly picked them that you checked one at random and found out that it was a boy. The process of selecting both kids was completely random.
I put a child in a room and ask you to guess the gender. What are your chances of being right?
I now say that I'm going to put a boy in the room too. Now what are your chances of being right about the genders?
quote:LOL, I had to head off to catch my commuter train home.
Originally posted by DonaldD:
Badvok, why did you quit on me above. You were soooo close![]()
quote:Nope that is not what was stated!
Originally posted by Pyrtolin:
We have one father and one boy, sure. but they were picked from the full set of families, some of which had no boys.
quote:Nope, I put one child in the room then added a boy. I didn't select two random children.
When you put two kids into the room, the possibility existed that you put two girls in as well. It's not till _after_ you randomly picked them that you checked one at random and found out that it was a boy. The process of selecting both kids was completely random.
quote:If you want to use balls then it is one blue ball and one ball that might be either blue or red in the bag. What is the probability there are two blue balls in the bag?
If I put my hand into a bag with an equal number of red and blue marbles and grab two at the same time, what's the chance that I pull out two blue ones together?
quote:Yep! The only math I have tried to correct is the erroneous elimination of an option from a probability space. This is, I think, the key cause of most people here misinterpreting the problem.
Originally posted by DonaldD:
You are arguing that the puzzle involves only a single unknown
quote:and rephrase it without using children:
John Doe comes up to you and says: "I have two kids. One of them is a boy. He was born on a Tuesday." What are the odds that he has two boys?
quote:Does this help at all?
I have a box that contains two chess pieces. One of them is white. It is also scratched. What are the odds that they are both white?
quote:
Originally posted by Badvok:
quote:Nope that is not what was stated!
Originally posted by Pyrtolin:
We have one father and one boy, sure. but they were picked from the full set of families, some of which had no boys.
quote:
quote:Nope, I put one child in the room then added a boy. I didn't select two random children.
When you put two kids into the room, the possibility existed that you put two girls in as well. It's not till _after_ you randomly picked them that you checked one at random and found out that it was a boy. The process of selecting both kids was completely random.
quote:[/qb][/quote]
quote:If you want to use balls then it is one blue ball and one ball that might be either blue or red in the bag. What is the probability there are two blue balls in the bag?
If I put my hand into a bag with an equal number of red and blue marbles and grab two at the same time, what's the chance that I pull out two blue ones together?
I put a stuffed toy and a child in a room and ask you to guess the child's gender. What are your chances?
I magically change the stuffed toy into a boy and tell you that is what I have done. What are your chances now?
quote:Where is the field? I can't see it! All I see is one father, one son and one other child.
John Doe comes up to you and says: "I have two kids. One of them is a boy. He was born on a Tuesday." What are the odds that he has two boys?
quote:I can assure you it's not a trap. The question is badly worded for the conclusion that you're meant to draw from it, but it is absolutely not a trap.
Badvok:
Unfortunately some have fallen into the trap cunningly presented to them.
quote:How would you go about answering this?
In front of you are two screens, standing side by side. You are told that there is one person behind both of them. You are further told that at least one of the two people behind the screeens is male. What is the probability that both of the people behind the screens are male?
quote:You're having a laugh aren't you?
Originally posted by Jordan:
How would you go about answering this?
quote:The field is that of all possible fathers. You're not in a magical world that only has this father or only fathers whose first child was a boy. Until he opened his mouth he could have had 100 daughters, for all you knew. When he does talk, he reveals that he's from the specific set of fathers with two children, and then narrows that to the set of fathers with two children, one of which is a boy. But all of those conditions were established randomly before he said a word- the only thing that has changed is the information that you have to work with.
Originally posted by Badvok:
Please! Someone please tell me that my English is not that bad that no one can understand me!
Why do people like Pyrtolin keep quoting me but then not understanding what I actually said?
There is no field from which a selection was made in the original statement:
quote:Where is the field? I can't see it! All I see is one father, one son and one other child.
John Doe comes up to you and says: "I have two kids. One of them is a boy. He was born on a Tuesday." What are the odds that he has two boys?
Please open my eyes! Or are we talking about an assumed field? As we all know Assume makes an ...
quote:Where did I ever say that the first child was a boy? Where does order come into it?
Originally posted by Pyrtolin:
The field is that of all possible fathers. You're not in a magical world that only has this father or only fathers whose first child was a boy.
quote:Full stop. What are the chances that the box contains two white pieces? Two black? A mix?
I have a box that contains two chess pieces.
quote:Noooo! You are changing the problem space again! The set of possibilities is only those where one of the pieces is white. There are no boxes where there is not at least one white piece because this is a stated fact! (Plus there is actually only one box.)
Originally posted by DonaldD:
Now, one of these men looks into his box at the pieces. By looking, he has not changed the pieces. He tells you that one of the pieces is white.
quote:I don't think there is much point do you?
Now, seriously, answer my previous post![]()
quote:Order comes into it because the order of the children introduces additional possibilities. The ambiguity of the order is exactly what's at key here and shifts the overall probability. The problem in all of your attempts to model the situation is explicitly that you don't account for the possibility of different orders or that he may have had two girls until he revealed order ambiguous information.
Originally posted by Badvok:
quote:Where did I ever say that the first child was a boy? Where does order come into it?
Originally posted by Pyrtolin:
The field is that of all possible fathers. You're not in a magical world that only has this father or only fathers whose first child was a boy.
quote:No- all fathers are in the problem space. He just happens to be the one that stepped forward and asked to be analyzed.
Where did I say he was the only father in the world? He is simply the only father in the problem space.
quote:Now that is a very big assumption! What did I say about assume?
Originally posted by Pyrtolin:
He just happens to be the one that stepped forward and asked to be analyzed.
quote:If you want to take order into account then you need to treat each child as a distinct entity and then there is a lot more than one combination for four boys - I'm too tired to work it all out now (time for me to head home) but I'm sure you can.
Originally posted by TomDavidson:
Ergo, we see one possible valid combination of four boys and four possible valid combinations of three boys and a girl.
quote:That's not an assumption. That's axiomatic. Using anything less than the general population when such a limit isn't explicitly stated in the problem is making an assumption.
Originally posted by Badvok:
quote:Now that is a very big assumption! What did I say about assume?
Originally posted by Pyrtolin:
He just happens to be the one that stepped forward and asked to be analyzed.
quote:A big assumption? It is indeed a big ASSUMPTION as I detailed in the first page of this thread, which you don't seem to have read.
Originally posted by Badvok:
quote:Now that is a very big assumption! What did I say about assume?
Originally posted by Pyrtolin:
He just happens to be the one that stepped forward and asked to be analyzed.
quote:This may be the source of your confusion. We don't particularly want to take order into account, but we have to take order into account. However, while there are a lot of possible permutations of four boys, we're only interested in the single combination of four boys; those permutations all collapse into one combination, because without any additional detail we can't tell any of the boys apart. They're all boys. The possible permuations, as written, are BBBB, BBBB, BBBB, and BBBB (etc.); they're all the same. The situation is different with one girl in the mix, since we can distinguish the permutation of BGBB from BBGB, BBBG, and GBBB.
If you want to take order into account then you need to treat each child as a distinct entity and then there is a lot more than one combination for four boys...
quote:Actually, yes. Your problem is one of semantics, and clarifying the initial statement is the only way to agree on what the puzzle actually means. You keep saying things like "Noooo! You are changing the problem space again!" but you refuse to clarify what you think the puzzle meant by "I have two kids. One of them is a boy".
I don't think there is much point do you?
quote:You are assuming that if at least one of the people is a male the announcer will tell you that there is a male. Without further information this is a bad assumption. If there is one male and one female, 50% of the time the announcer will choose male and 50% of the time the announcer will choose female.
Let's see exactly how you're approaching this. I'm going to try and reword the question a little more clearly:
quote:How would you go about answering this?
In front of you are two screens, standing side by side. You are told that there is one person behind both of them. You are further told that at least one of the two people behind the screeens is male. What is the probability that both of the people behind the screens are male?
quote:Of the 50 families with a boy and a girl, what are the odds of a father choosing to tell you he has a boy?
Originally posted by threads:
Badvok:
There are 100 two-child families in the room. The world is naturally perfect so 25 of the families have 2 boys, 25 of the families have 2 girls, and 50 of the families have a boy and a girl. A father comes up to you and says "at least one of my children is a boy". What is the probability that he has two boys?
quote:The intial problem is not ambiguous. The answer is 1/2. In order for the answer 1/3 (or 13/27) we have to assume the father would only tell us the gender if he had a boy and only tell us the day of the week if it happened to be Tuesday.
So yes, with the initial phrasing it's very ambiguous, but you were given the unambiguous phrasing in which the dad volunteers no other information except "two children" -- and you still believed it was 1/2.
quote:Sorta. It depends on the guy's reason for volunteering the information. He may have volunteered it for reasons that resolve to the same 33% possibility (e.g. asking army recruitment information, so all the fathers with atleast one boy would have to go there, while fathers with only girls wouldn't).
If the information is volunteered the odds of the second kid being a boy is 50/50
quote:You should read Aris' discussion of this point. Relevant post:
Originally posted by Ciasiab:
quote:Of the 50 families with a boy and a girl, what are the odds of a father choosing to tell you he has a boy?
Originally posted by threads:
Badvok:
There are 100 two-child families in the room. The world is naturally perfect so 25 of the families have 2 boys, 25 of the families have 2 girls, and 50 of the families have a boy and a girl. A father comes up to you and says "at least one of my children is a boy". What is the probability that he has two boys?
quote:
Originally posted by Aris Katsaris:
Let me put it in another way.
Possibilities for two children:
G/G 1/4
G/B 1/4
B/G 1/4
B/B 1/4
--
If guy decides to reveal one of these genders randomly, possibilities now become
G/G (reveals G) 1/4
G/B (reveals 1st: G) 1/8
G/B (reveals 2nd: B) 1/8
B/G (reveals 1st: B) 1/8
B/G (reveals 2nd: G) 1/8
B/B (reveals B) 1/4
Now if we know "B" was revealed, this corresponds to 1/8 + 1/8 + 1/4 = 50%. Out of these, it's even odds that the other one was a boy, or that the other one was a girl.
So if the guy randomly selected the kid whose gender he'd reveal, it's even odds that the other kid is either gender. 50% says common sense, and 50% it indeed is.
--
HOWEVER if the guy thinks: I will NOT mention there's a girl, but I will only reveal if there exists a boy. The possibilities become:
G/G (mentions no information) 1/4
G/B (mentions there's a B) 1/4
B/G (mentions there's a B) 1/4
B/B (mentions there's a B) 1/4
Now, knowing he revealed it was a boy, there only 33.3% chances that the other kid is a boy too, and 66.7% chances that the other kid is a girl.
--
AND if the guy thinks: I will mention ALL my boys, but none of my girls. The possibilities become:
G/G (mentions no information) 1/4
G/B (mentions there's a B) 1/4
B/G (mentions there's a B) 1/4
B/B (mentions there's two boys) 1/4
Now, with the knowledge he mentioned only *one* B for certain, we can be 100% sure that the other kid is a girl -- because he'd have mentioned two boys if B/B was the reality.
--
That's what I mean when I say motivation matters. WHY did he reveal the particular gender? Was he randomly picking a kid, or was he choosing that gender for some reason?
quote:You have to project additional reasons to get there. You have to add an assumption about army recruitment or a sexist dad.
Sorta. It depends on the guy's reason for volunteering the information. He may have volunteered it for reasons that resolve to the same 33% possibility (e.g. asking army recruitment information, so all the fathers with atleast one boy would have to go there, while fathers with only girls wouldn't).
quote:The original phrasing is ambiguous exactly because we don't know if these assumptions are right or wrong. A Spartan soldier volunteering to be among the 300 would want at least one son to carry the family line. The existence or non-existence of girls wouldn't matter to him.
"The initial problem is not ambiguous. The answer is 1/2. In order for the answer 1/3 (or 13/32) we have to assume the father would only tell us the gender if he had a boy and only tell us the day of the week if it happened to be Tuesday. "
quote:Aris' post proves my point. Unless you assume the father is not providing random infomation (i.e. he's at an army recruiting station), the problem as stated resolves to 1/2.
You should read Aris' discussion of this point. Relevant post:
quote:Sure, but "I've randomly picked a kid of mine whose gender I'm telling you" is also an assumption, as you don't know that.
"You have to project additional reasons to get there."
quote:Most problems can be ambiguous. You can read it as random, or you can add assumptions in your head. The only way to solve the problem is to assume it is random. If you add another assumption you are adding to the problem (like saying you happen to be at a military recruitment center). If you assume (another assumption I know) that all the information is contained in the problem statement, then I don't see how not to conclude he is choosing a kid at random.
Originally posted by threads:
Ciasiab, you are projecting just as much by assuming that the father is randomly choosing which kid to talk about. That's not stated anywhere in the initial problem. Just agree that it's ambiguous and let it be.
quote:Edit to add: How about we call it ambiguous and leave it at that.
Originally posted by Ciasiab:
quote:Most problems can be ambiguous. You can read it as random, or you can add assumptions in your head. The only way to solve the problem is to assume it is random. If you add another assumption you are adding to the problem (like saying you happen to be at a military recruitment center). If you assume (another assumption I know) that all the information is contained in the problem statement, then I don't see how not to conclude he is choosing a kid at random.
Originally posted by threads:
Ciasiab, you are projecting just as much by assuming that the father is randomly choosing which kid to talk about. That's not stated anywhere in the initial problem. Just agree that it's ambiguous and let it be.
quote:Eh. What if the randomness is that he randomly chooses a gender, and then mentions its existence or non-existence?
You can read it as random, or you can add assumptions in your head. The only way to solve the problem is to assume it is random.
quote:You make a very excellent point.
Eh. What if the randomness is that he randomly chooses a gender, and then mentions its existence or non-existence?
quote:No, Badvok; I'm trying to make an opening—and something of a break from the prior discussion—with what I hope is a somewhat cleaner (and clearer) example.
Badvok:
You're having a laugh aren't you?
quote:Badvok - This post by Threads seems to be the best way to phrase the question so as to be understood. Does it make more sense when it is put this way?
There are 100 two-child families in the room. The world is naturally perfect so 25 of the families have 2 boys, 25 of the families have 2 girls, and 50 of the families have a boy and a girl. A father comes up to you and says "at least one of my children is a boy". What is the probability that he has two boys?
quote:LOL, no I don't think I KEPT insisting that phrasing meant the probability was 1/2. I admit that I did mistakenly respond that way once though.
Originally posted by Aris Katsaris:
"John Doe: I have two children.
Me: Is at least one a boy?
John Doe: Yes."
And you KEPT insisting that the probability for two sons is 1/2 even with this rephrasing.
quote:Even still, his children come in the following permutations:
To me he is one man, he has one son and there is one child we do not know the gender of.
quote:No drama, Badvok.
Badvok:
You are obviously not talking about the original statement and I am just confusing things by not accepting your assumptions about it.
quote:*I*, for one, wasn't making assumptions, I rephrased the question so that no assumptions needed be made, and I detailed in the first page of this thread how different assumptions (that other people were making) affect the probability.
You are obviously not talking about the original statement and I am just confusing things by not accepting your assumptions about it.
quote:No, he is one man, and he has two children whose gender we don't know of -- with the knowledge that atleast one of them is a son affecting the corresponding probabilities for each child being a son.
"To me he is one man, he has one son and there is one child we do not know the gender of."
quote:
Originally posted by Badvok:
Your numbers are not wrong in that sense.
1/3 of the 75 fathers (from a population of 100) who have at least one son have two sons.
quote:I interpreted that, combined with your earlier comparison of bad arithmetic, as meaning that you didn't believe the same logic would apply in the case of one father about whom we know only that he has at least one son. Was I wrong?
Badvok:
But you are just solving the wrong problem and hence using the wrong numbers for the problem. We have ONE father, ONE son, and ONE other child who's gender we do not know.
quote:All I see is one father, one son and one other child.
John Doe comes up to you and says: "I have two kids. One of them is a boy. He was born on a Tuesday." What are the odds that he has two boys?
quote:I'm unimpressed enough by the original phrasing of the question that by now I'm willing to grant anyone's particular interpretation rather than bothering to argue. As I said on the first page it's pretty ambiguous.
Badvok:
Why is there only one other child and hence only a single unknown? The statement 'HE was born on a Tuesday." is not totally irrelevant but identifies one of the two children because of the pronoun used!
quote:Completely agreed with this. As the pronoun specifically identifies the child, the odds are at 50%.
Why is there only one other child and hence only a single unknown? The statement 'HE was born on a Tuesday." is not totally irrelevant but identifies one of the two children because of the pronoun used!
quote:BTW, Badvok
If he had said "one is a boy born on Tuesday" that would be different because there is no pronoun and hence no identification of an individual.
The "Tuesday" value itself is irrelevant because the question is only about gender.
quote:I don't think I agree with your interpretation of my statement, here. I suggested something being scratched as not equivalent to a child being born on Tuesday not because it identifies the item (it does not necessarily, any more than does a Tuesday birth) but rather because the probability of a scratch is indeterminate.
Thanks to DonaldD for questioning my inclusion of "It was scratched." in my alternate version - it is not irrelevant it is key to reducing the problem to a single unknown.
quote:Aside from being the only time I recall seeing the word "specificity" used to refer to anything other than CSS selectors, that's an admirably clear summary of the principle at work, Aris.
Aris Katsaris:
The more specificity about the child in our probing question the more the probability goes from 33% to 50%. For example.
Father: I have two children.
Me: Was at least one of them a boy that is the current Governor of Alabama?
Father: Yes.
This is 50% again since the boy is specifically identified by the fact only one governor of Alabama can exist at a time.