Another math problem

[Nathelion]

I thought I should jump on the bandwagon and provide some additional amusement for those of us who enjoy the math threads. This thread is more fun than business, I already know the answer to the problem in question, so don't feel compelled to post just to be nice/help out.

For the purposes of this discussion, there is always a 50% chance that a woman giving birth will have a boy and a 50% chance of her having a girl (this is not true in reality).

Imagine that you are going to a party. You arrive, and a man and two boys greet you. The man says "These two boys are my sons, but I have three children. The third child is in the back yard playing." You can't see the back yard from where you are standing. You have no prior knowledge of this man's family.

What is the probability that the child playing in the back yard is a girl?

 

[pcy]

75%


Peter

 

[Nathelion]

You, sir, are ruining all my fun.

I just wanted to watch people mess up and not get it at all for weeks and weeks while I smugly smile into my beard and laugh and feel superior, and you just pop the answer out like that? Horrible manners:evil:

 

[pcy]

Hi,


Sorry... but even if they know the correct answer (and who's to say that 75% is correct) they can still discuss why it right or wrong and get confused.

Alternatively ask a difficult question, and I endevour to post a plausible but incorrect answer.



Peter

 

[stupidkid]

Conditional probability. We are given that there are at least 2 boys and need to find the probability that the last child is a girl.

So P(A|B) = P(A int B)/P(B)

P(B) = probability there are at least 2 boys (BBB GBB BGB BBG), which is 4 * 1/8 = 1/2

P(A int B) = probability of 1 girl and 2 boys (GBB BGB BBG), which is (3 choose 2) * 1/8 = 3/8

Therefore (3/8)/(1/2) = 3/4.

Edit: I like probability problems, if you have another one, please post it.

 

[pcy]

Actually... that's not absolutley correct. My answer of 75% is wrong because the fact that a man is known to have at least two male children out of 3 must impact the probablitly of the sex of the third child.



Peter

 

[stupidkid]

No its correct. This problem is conditional probability. He has at least 2 sons, which is the condition.

 

[pcy]

Yes - the point I'm making is that the fact that he has two sons out of three children increases the chances he has some medical condition that pre-disposes him to have sons rather thatn daughters.

The original assumption that the chances of a male and female child are both 50% is an unconditional probability. As soon as you know that a particular man has at least two sons out of three children the probability of other children being male or female becomes conditional on the piece of information also.

I'm sure it's very close to 75%...



Peter

 

[MikeyLSU]

now, why would knowing that they have 2 boys already have any impact on the probability that the last one is a girl.

The question only asks what is the gender of that particular child, it does not ask what is the probability that all 3 will be boys or 2 boys and 1 girl.

unless worded differently, I still see it as 50%

For instance, the child in the back could be the oldest child or the youngest(not that it really matters) though.

EDIT to make point clearer:

The question could just as easily be worded that you meet a guy with 2 sons and a pregnant wife. What is the probability that the newborn will be a girl. It is still 50/50...previous children do not change the outcome of the 3rd child.

 

[pcy]

Oh but they do.

Not by much, bu the fact that a man has two sons out of two children (the man determines the sex of the child) clearly increases the probability that he is predisposed to father sons rather than daughters because of some medical (or other?) condition.

Suppose you ran in to a man who had 1000 children, all boys. The chances that this has happened "just by chance" and the actual probability of each child being a boy was 50% is 1/2*100 - very small indded, so you would have to consider the possibility that he only produces Y sperm (or is it X?).

The point here is that as soon as we know something abut the genders of the man's children it provides additional information which has some (however small) effect on the probable sex of further children.




Peter

 

[MikeyLSU]

but you are saying it would be 75% a girl because of the overall probability of having 3 boys vs. 2 boys and a girl. I'm saying the unknown is separate without any other knowledge because you don't know if it is the first second or third child. And we are assuming a 50% chance of every child being a girl(from OP)

 

[Thyme]

Originally posted by: MikeyLSU
but you are saying it would be 75% a girl because of the overall probability of having 3 boys vs. 2 boys and a girl. I'm saying the unknown is separate without any other knowledge because you don't know if it is the first second or third child. And we are assuming a 50% chance of every child being a girl(from OP)

I agree with that. If I flip two coins in front of you and they're heads, and I flipped a coin that's away from you, it's still a 50% chance that the one away from you is also a head.

Alternatively, I'd say about 95% based on the wording--these two boys are my sons implies to some degree they are his only sons; it'd be more natural to say "these two boys are my sons, but I have three boys. The third is playing outside"

 

[pcy]

Originally posted by: MikeyLSU
but you are saying it would be 75% a girl because of the overall probability of having 3 boys vs. 2 boys and a girl. I'm saying the unknown is separate without any other knowledge because you don't know if it is the first second or third child. And we are assuming a 50% chance of every child being a girl(from OP)

The order of the children's birth is irrelevant because we don't know it. Conditional probability is contingent on known information. We are no longer assuming a 50/50 split becaue we alrady know the answer tho that question, and because it is the unconditional probability and we have additional information.


The question is: does the fact that a man has at least two male children out of three change (however slightly) the probability of the third one also being male.

If you think not, tell me if you think that a man with at least 999 male children out of 1000 wouls possibly/probably have some medical condition that predisposed hin to fathering male children?



Peter

 

[pcy]

Originally posted by: Thyme

Originally posted by: MikeyLSU
but you are saying it would be 75% a girl because of the overall probability of having 3 boys vs. 2 boys and a girl. I'm saying the unknown is separate without any other knowledge because you don't know if it is the first second or third child. And we are assuming a 50% chance of every child being a girl(from OP)

I agree with that. If I flip two coins in front of you and they're heads, and I flipped a coin that's away from you, it's still a 50% chance that the one away from you is also a head.

You are asserting by your example that the chances of the third coin coming down heads or tails is independant of the other two coins/results.

A better analogy would be one coin flipped three times: same father, same coin. The more times the coin turns up heads with no tails, the more likely I am to think it's biased in some way...




Peter

 

[Peter]

Basic rules of probabilties 101, m'learned friends.

Independent "drawings" is what we have here.

50% for /every/ child born. Regardless of what previous children turned out to be, the next one will still be a 50/50.

There is no dependency from previous events here. This would be different if the husband had 50/50 boy/girl sperm and was running low on the former because he's already produced two boys. But we all know it doesn't work like that, don't we?

 

[Born2bwire]

Originally posted by: Peter
Basic rules of probabilties 101, m'learned friends.

Independent "drawings" is what we have here.

50% for /every/ child born. Regardless of what previous children turned out to be, the next one will still be a 50/50.

There is no dependency from previous events here. This would be different if the husband had 50/50 boy/girl sperm and was running low on the former because he's already produced two boys. But we all know it doesn't work like that, don't we?

The final probability does matter if we do not know the relative ages of the children. There's only one way to have three boys BBB, but three ways to have two boys and a girl (BBG, BGB, GBB) with each "way" having the same probability. All of this was discussed previously in OT.

Text

They even addressed the idea that PCY mentioned that if you had two boys previously then you might be physically predisposed to one sex over the other, which isn't really true (unless you feel that a change of 2% is significant but without knowing the details of the statistics it's hard to say).

 

[MikeyLSU]

Originally posted by: Born2bwire

Originally posted by: Peter
Basic rules of probabilties 101, m'learned friends.

Independent "drawings" is what we have here.

50% for /every/ child born. Regardless of what previous children turned out to be, the next one will still be a 50/50.

There is no dependency from previous events here. This would be different if the husband had 50/50 boy/girl sperm and was running low on the former because he's already produced two boys. But we all know it doesn't work like that, don't we?

The final probability does matter if we do not know the relative ages of the children. There's only one way to have three boys BBB, but three ways to have two boys and a girl (BBG, BGB, GBB) with each "way" having the same probability. All of this was discussed previously in OT.

Text

They even addressed the idea that PCY mentioned that if you had two boys previously then you might be physically predisposed to one sex over the other, which isn't really true (unless you feel that a change of 2% is significant but without knowing the details of the statistics it's hard to say).

While that could or could not be true, it is all assumptions. Here is this taken from the OP in the question:

"For the purposes of this discussion, there is always a 50% chance that a woman giving birth will have a boy and a 50% chance of her having a girl (this is not true in reality). "

Therefore, the probability the other child is a girl is still 50% becuase nothing else matters if there is a 50/50 chance of it being a girl.

 

[pcy]

Hi,

Originally posted by: Peter
Basic rules of probabilties 101, m'learned friends.

Independent "drawings" is what we have here.

50% for /every/ child born. Regardless of what previous children turned out to be, the next one will still be a 50/50.

There is no dependency from previous events here. This would be different if the husband had 50/50 boy/girl sperm and was running low on the former because he's already produced two boys. But we all know it doesn't work like that, don't we?

Independant drawing yes - but they are linked as they share the father.

The question is whether the conditional value of the prbability of the sex of the child remains constant at the nominal 50/50 for the unconditional value once we know that the father has at least two male cildren out of a total of three.


Tbe original question did say we should assume that this value does remain constant, but it beggars belief that Nathelion intended the conditional probability to remain at 50/50. Suppose we know that all there children were boys then the probability of the firstborn (given this additional information) being a boy is clearly 100%, not 50%.

The point is simply that once we have additional information the probailities of events change. All I'm saying is that the probability of a man fathering boys rather than girls does change one we know the sexes of his children, because it impacts our assessment of his relative X and Y sperm count, which will in turn affect the probable sex of other children.



Peter

 

[Born2bwire]

Originally posted by: MikeyLSU

Originally posted by: Born2bwire

Originally posted by: Peter
Basic rules of probabilties 101, m'learned friends.

Independent "drawings" is what we have here.

50% for /every/ child born. Regardless of what previous children turned out to be, the next one will still be a 50/50.

There is no dependency from previous events here. This would be different if the husband had 50/50 boy/girl sperm and was running low on the former because he's already produced two boys. But we all know it doesn't work like that, don't we?

The final probability does matter if we do not know the relative ages of the children. There's only one way to have three boys BBB, but three ways to have two boys and a girl (BBG, BGB, GBB) with each "way" having the same probability. All of this was discussed previously in OT.

Text

They even addressed the idea that PCY mentioned that if you had two boys previously then you might be physically predisposed to one sex over the other, which isn't really true (unless you feel that a change of 2% is significant but without knowing the details of the statistics it's hard to say).

While that could or could not be true, it is all assumptions. Here is this taken from the OP in the question:

"For the purposes of this discussion, there is always a 50% chance that a woman giving birth will have a boy and a 50% chance of her having a girl (this is not true in reality). "

Therefore, the probability the other child is a girl is still 50% becuase nothing else matters if there is a 50/50 chance of it being a girl.

But you are assuming the condition that the third child is of a specific age relative to the boys. Without any assumptions, all we know is that we have a father with at least two boys. Of the possible groups of two boys, he could be BBB, GBB, BGB, or BBG in terms of their relative ages. We do not know which group that the third child places the father into. His third child could be the eldest, youngest, or middle child. If the third child is a boy, then the relative age does not matter, but if the child is a girl, then there are three distinct groups that will give rise to the same outcome.

Still the simplest way to solve this is to just draw out a tree of all the possible child combinations. There are 8 possible ones, but only 4 where there are at least two boys and of those only 1 where you have three boys. So you you have 3/4 probability that the guy has a daughter since the probability to have a girl or a boy is the same.

Here's another link about the Marilyn Vos Savant article. Her problem focuses on two children and the main point of contention is the fact that the problem statement is ambiguous. Fortunately, the OP's problem statement is not ambiguous. Still, they treat the concept of relative ages as a determining factor.

Text

 

[Nathelion]

As to the whole debate about whether having two male children would indicate that the father is more likely to have more male children, in the problem as intended this is not the case. It is an interesting discussion of course, but has less to do with math and more to do with statistics (of the real world kind).

Alternatively, I'd say about 95% based on the wording--these two boys are my sons implies to some degree they are his only sons; it'd be more natural to say "these two boys are my sons, but I have three boys. The third is playing outside"

The problem in its intended version is not about grammar. Grammar is also fun though, almost as much fun as math

 

[Nathelion]

Originally posted by: pcy
Hi,


Sorry... but even if they know the correct answer (and who's to say that 75% is correct) they can still discuss why it right or wrong and get confused.

Alternatively ask a difficult question, and I endevour to post a plausible but incorrect answer.



Peter

Haha that would be fun.

 

[Thyme]

Originally posted by: Nathelion
As to the whole debate about whether having two male children would indicate that the father is more likely to have more male children, in the problem as intended this is not the case. It is an interesting discussion of course, but has less to do with math and more to do with statistics (of the real world kind).

Alternatively, I'd say about 95% based on the wording--these two boys are my sons implies to some degree they are his only sons; it'd be more natural to say "these two boys are my sons, but I have three boys. The third is playing outside"

The problem in its intended version is not about grammar. Grammar is also fun though, almost as much fun as math

That's another assumption that wasn't stated. Since the problem involved a man *talking* and not just something that happened to be or not be, it makes sense to look at the probabilities of a random person speaking in one way versus another. I submit that speaking patterns are more of a factor than a given man's previous offspring.

But grammar aside, we assumed that there is always a a 50% chance of a woman giving birth to a boy. That means it doesn't matter about the man because it's always independent. So the chance of any given child being a boy is 50% and so the chance that the unknown kid is a boy is also 50%. The assumption would not hold if it were not independent.

 

[imported_Finny]

Ugh, I hate statistics...

Anyone who thinks this has to do with the chance of the outcome being altered by previous births is mistaken. That's not where the trick lies.

The trick comes in the interpretation. Normally, one sees "what's the chance this child is a boy or girl?", and realizes, "The outcome is independent of other outcomes, so it must be 50%!". However, that's not what the question is asking; what it's asking is, what's the chance taking into account that there are also two boys; more specifically, what's the chance of a Boy-Boy-Girl series of births, since we know two of them are boys (Of course, the fact that we don't know what order the births came in also puts a damper on things).

If you want to get into details about it, you could check out geometric distribution on wikipedia. In any case, the basic statistical answer is 75%, not taking into account any other cirmumstances, like pcy's idea of a specific gender of one child affecting that of the next (Which is shot down to begin with by the problem's statement of there always being a 50/50 chance of boy/girl)

 

[JJChicken]

Originally posted by: Finny
Ugh, I hate statistics...

Anyone who thinks this has to do with the chance of the outcome being altered by previous births is mistaken. That's not where the trick lies.

The trick comes in the interpretation. Normally, one sees "what's the chance this child is a boy or girl?", and realizes, "The outcome is independent of other outcomes, so it must be 50%!". However, that's not what the question is asking; what it's asking is, what's the chance taking into account that there are also two boys; more specifically, what's the chance of a Boy-Boy-Girl series of births, since we know two of them are boys (Of course, the fact that we don't know what order the births came in also puts a damper on things).

If you want to get into details about it, you could check out geometric distribution on wikipedia. In any case, the basic statistical answer is 75%, not taking into account any other cirmumstances, like pcy's idea of a specific gender of one child affecting that of the next (Which is shot down to begin with by the problem's statement of there always being a 50/50 chance of boy/girl)

nope 75% is correct. i remember all of us going "NO! WTF 50% FTW!!" in stats class last year, but 75% is the right answer. im not smart enough to remember why tho..

 

[imported_Finny]

That's what I said at the end, the basic answer is indeed 75%. Someone else explained it nicely above, so I don't feel like repeating it, nor do I feel like getting into the deeper statistical analyzation, which I did more than enough of last semester