## Given that you have a brother...

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LSK
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### Given that you have a brother...

...is it more likely that you're male or female?

Assumptions: There's an exactly 50% chance of being born male or female. Parents have a finite number of children.
Last edited by LSK on Tue Apr 13, 2010 6:58 pm UTC, edited 2 times in total.

JBJ
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### Re: Given that you have a brother...

Spoiler:
Neither is more likely. Are we assuming you and your brother are the only two children?

Code: Select all

P1     P2--------------Male   MaleMale   FemaleFemale MaleFemale Female

If you are P1, then what are the options when P2 is male?
1 Male, 1 Female

If you are P2, then what are the options when P1 is male?
1 Male, 1 Female
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ttnarg
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### Re: Given that you have a brother...

LSK wrote:...is it more likely that you're male or female?

Assumptions: There's an exactly 50% chance of being born male or female. Parents have a finite number of children.

I'm not 100% sure but I think....
Spoiler:
I think the anser is you have a 50:50 chance of being male or female but I'm not 100% sure.

but the reson I'm not sure is if there where 100 children 50 of them male and 50 of them female then and you had a brother then there are only 49 males you could be and 50 females. but I dont think thats how it work I think it dose not matter if you have a brother or not. each person starts with a 50:50 change of being male or female. having a brother gives infomation about a group(1 or more) of people gender but that group dose not incuilde you.

Here is another question with I think a diffent anser: If you got all the people who have brothers is it more likey that they are male or female?

Goldstein
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### Re: Given that you have a brother...

Spoiler:
Imagine a family with n children, where n >= 1.

If there are two or more boys in the family, every child has a brother.

(1/2)^n families of this type consist of children who are all girls, and so none of these n girls have brothers.

n*(1/2)^n families of this type consist of children among whom exactly one is a boy, and so everyone but one boy has a brother.

Out of 2^n families with n children, we have the same expected number of boys and girls, by virtue of symmetry. In this population of families, we have an expected one family in which n girls have no brothers, and n families in which one boy has no brothers. All other children have brothers. Thus, we have the same number of boys and girls who have brothers in a family of size n, for all n>=1.

And so given that I have a brother and my family contains n children (n being at least 2 in this case), it's equally likely that I'm a boy or a girl.
Spoiler:
I'm actually a boy though.
Last edited by Goldstein on Tue Apr 13, 2010 10:16 pm UTC, edited 1 time in total.
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Hix
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### Re: Given that you have a brother...

ttnarg wrote:Here is another question with I think a diffent anser: If you got all the people who have brothers is it more likey that they are male or female?

No, this wouldn't have a different answer. This is practically the definition of the conditional probability meant by "given that you have a brother, is it more likely that they are male or female".

Anyway,the probability will depend on society-wide family distribution tendencies. For example, if no family stops having kids until 2 males have been born, then precisely 50% of people with brothers are male. But if every family always stops having kids after their first male, then 0% of people with brothers are male. I can't think of any scenario consistent with a 50/50 birth rate allows more than 50% of "brothered" individuals to be male. Possibly it's provable.

Goldstein
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### Re: Given that you have a brother...

Hix wrote:if every family always stops having kids after their first male, then 0% of people with brothers are male.

This is a good point, the problem as stated didn't rule this out. I have made the additional assumption that all parents are blind!
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Hix
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### Re: Given that you have a brother...

Hmm... Just thought of a way it could be a greater than 50% chance that you are male. Suppose no family ever stops at exactly 1 male child, but that some families are satisfied and stop when they have only females. Then the collection of people with brothers would include every male (males with brothers therefore accounting for 50% of the overall population), but wouldn't include every female (females with brothers therefore accounting for less than 50% of the overall population). So the conditional probability in question would end up being over 50%. Probably, the strategy for maximizing this conditional probability would be for a family to stop immediately if the first child is female (don't risk letting her get a brother), otherwise to stop immediately after the second male (If you've got a male without a brother, it's always worth it to try to get him a younger brother (who will automatically have an older brother!). But once you've got 2 males, additional children only serve to move the conditional probability back toward 50%)

xkcdfan
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### Re: Given that you have a brother...

But I don't have a brother.

dawolf
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### Re: Given that you have a brother...

Hix wrote:Hmm... Just thought of a way it could be a greater than 50% chance that you are male. Suppose no family ever stops at exactly 1 male child, but that some families are satisfied and stop when they have only females. Then the collection of people with brothers would include every male (males with brothers therefore accounting for 50% of the overall population), but wouldn't include every female (females with brothers therefore accounting for less than 50% of the overall population). So the conditional probability in question would end up being over 50%.....

It would be 66% chance that someone with a brother is male.

female
male female
male male

are the options. 1 female and two males have brothers.

douglasm
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### Re: Given that you have a brother...

dawolf wrote:
Hix wrote:Hmm... Just thought of a way it could be a greater than 50% chance that you are male. Suppose no family ever stops at exactly 1 male child, but that some families are satisfied and stop when they have only females. Then the collection of people with brothers would include every male (males with brothers therefore accounting for 50% of the overall population), but wouldn't include every female (females with brothers therefore accounting for less than 50% of the overall population). So the conditional probability in question would end up being over 50%.....

It would be 66% chance that someone with a brother is male.

female
male female
male male

are the options. 1 female and two males have brothers.

You're forgetting "female male". Yes, order matters. Probability is the same for all combinations only if you distinguish different orderings as different combinations.

rigwarl
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### Re: Given that you have a brother...

douglasm wrote:You're forgetting "female male". Yes, order matters. Probability is the same for all combinations only if you distinguish different orderings as different combinations.

"female male" is not possible in the situation dawolf is addressing (family stops if they have a daughter). Although I believe the 66% answer may require the additional constraint of only 2 kids maximum.

For the problem in the original post, the answer is:

Spoiler:
50/50. This differs from the "2 puppies" problem that has been on the front page for awhile in that you know exactly which one of the two is the male rather than just that at least one is.

(replace two with n for more complex version)

douglasm
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### Re: Given that you have a brother...

rigwarl wrote:
douglasm wrote:You're forgetting "female male". Yes, order matters. Probability is the same for all combinations only if you distinguish different orderings as different combinations.

"female male" is not possible in the situation dawolf is addressing (family stops if they have a daughter). Although I believe the 66% answer may require the additional constraint of only 2 kids maximum.

Oops, didn't notice that part. That would indeed change things.

rigwarl wrote:For the problem in the original post, the answer is:

Spoiler:
50/50. This differs from the "2 puppies" problem that has been on the front page for awhile in that you know exactly which one of the two is the male rather than just that at least one is.

(replace two with n for more complex version)

Spoiler:
That's one way to look at it, but I'm not sure it's correct. The difference I thought of is that male/male counts double for this problem, but only counts once for the puppies problem. Here, the probability that matters is per person. For the 2 puppies problem, the probability that matters is per store.

Either way, 50% is the correct answer if family size is independent of child gender.

Token
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### Re: Given that you have a brother...

Hix wrote:Hmm... Just thought of a way it could be a greater than 50% chance that you are male. Suppose no family ever stops at exactly 1 male child, but that some families are satisfied and stop when they have only females. Then the collection of people with brothers would include every male (males with brothers therefore accounting for 50% of the overall population), but wouldn't include every female (females with brothers therefore accounting for less than 50% of the overall population). So the conditional probability in question would end up being over 50%. Probably, the strategy for maximizing this conditional probability would be for a family to stop immediately if the first child is female (don't risk letting her get a brother), otherwise to stop immediately after the second male (If you've got a male without a brother, it's always worth it to try to get him a younger brother (who will automatically have an older brother!). But once you've got 2 males, additional children only serve to move the conditional probability back toward 50%)

This example assumes that all families have reached a point at which they no longer wish to have children. A necessary consequence of this is the allowance for a family to have an arbitrarily large number of children, which kind of conflicts with biological restrictions.
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### Re: Given that you have a brother...

Token wrote:
Hix wrote:Hmm... Just thought of a way it could be a greater than 50% chance that you are male. Suppose no family ever stops at exactly 1 male child, but that some families are satisfied and stop when they have only females. Then the collection of people with brothers would include every male (males with brothers therefore accounting for 50% of the overall population), but wouldn't include every female (females with brothers therefore accounting for less than 50% of the overall population). So the conditional probability in question would end up being over 50%. Probably, the strategy for maximizing this conditional probability would be for a family to stop immediately if the first child is female (don't risk letting her get a brother), otherwise to stop immediately after the second male (If you've got a male without a brother, it's always worth it to try to get him a younger brother (who will automatically have an older brother!). But once you've got 2 males, additional children only serve to move the conditional probability back toward 50%)

This example assumes that all families have reached a point at which they no longer wish to have children. A necessary consequence of this is the allowance for a family to have an arbitrarily large number of children, which kind of conflicts with biological restrictions.

Actually that doesn't matter.

Suppose everyone stops after their first child if it's a girl, and otherwise has two children. Then half of all families who have had time to have two children have one girl, one quarter of all those families have two boys, and one quarter of all those families have a girl and a boy. So if there are n such families, you end up with n/2 girls with no brother, n/4 girls with a brother, n/4 boys with no brother, and n/2 boys with a brother. Meaning that among people with brothers, 2/3 of them are boys. (Of course, taking into account the families that haven't had time to have two children makes no difference to the ratio we are interested, since none of those families contain children with brothers.)

In fact, suppose everyone follows the strategy "stop having children if your first child is a girl, or if you have two boys." Let's look at the subgroup of these families who have had time to have k children. Then half of all these families will have one girl, and the other girls will all have brothers, while 1/2k families will only have one boy. Furthermore, each family has a 50% chance of having one child, a 25% chance of two, and so on, up to 1/2k-1 chance of having k-1 children and 1/2k-1 chance of having k children. So the expected number of children will be $$\frac12+\frac24+\frac38+...+\frac{k-1}{2^{k-1}}+\frac{k}{2^{k-1}}=\sum_{i=1}^\infty \frac{i}{2^i}-\sum_{i=k}^\infty \frac{i}{2^i}+\frac{k}{2^{k-1}}=2-\sum_{i=1}^\infty \frac{i+k-1}{2^{i+k-1}}+\frac{k}{2^{k-1}}$$ $$=2-\frac{1}{2^{k-1}}\left(\sum_{i=1}^\infty \frac{i}{2^{i}}+\sum_{i=1}^\infty \frac{k-1}{2^{i}} \right)+\frac{k}{2^{k-1}}=2-\frac{1}{2^{k-1}}(2+k-1)+\frac{k}{2^{k-1}}=2-\frac{1}{2^{k-1}}.$$
Therefore, on average per family in this group, there will be 1-1/2k boys (with 1/2k brotherless boys) and 1-1/2k girls (with 1/2 brotherless girls), so among children with brothers, there will be 1-2/2k boys and 1/2-1/2k girls, so 2/3 of children with brothers will be boys. In other words, among every group of families, 2/3 of children with brothers will be boys, so if you have a brother, you will have a $$66.\overline{6}\%$$ chance of being a boy. This is the same figure we would get if every family had finished having children.
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Pesto
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### Re: Given that you have a brother...

Spoiler:
Your chances of being either a male or female are equal. The question is a red herring, as the gender of your siblings have no bearing on your own gender.

Edit: A more rigorous solution.

Code: Select all

People with brothers shown in parentheses| P1  P2|+-------+| F   F ||(F)  M || M  (F)||(M) (M)|Of the four people who have brothers, two are male, two are female.

Looking at larger families, this solution abstracts out to be a general solution.

Spoiler:
Given that I actually do have a brother, it's 100% likely that I am male, because I am male.

Me321
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### Re: Given that you have a brother...

About 51.456310679611650485436893203883% chance that I am male.

http://en.wikipedia.org/wiki/Sex_ratio

bane2571
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### Re: Given that you have a brother...

Spoiler:
In any N people, 50% are male

If 1 male is my brother (IE not me) then I have a slightly decreased chance of being male.

We are talking about a pre-existing population so information about people that aren't are male helps determines my probability of being male

Where is the fallacy in that? Everyone seems to be thinking differently.

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### Re: Given that you have a brother...

bane2571 wrote:In any N people, 50% are male

If 1 male is my brother (IE not me) then I have a slightly decreased chance of being male.

We are talking about a pre-existing population so information about people that aren't are male helps determines my probability of being male.

Where is the fallacy in that? Everyone seems to be thinking differently.

It's not true that in N people, 50% are male. What is (roughly) true is that each baby, at conception, has a roughly 50% chance of being a male baby. If we are interested in knowing Alex's sex, knowing the sex of people other Alex doesn't give us any new information about the Alex's sex. The idea that it does is essentially the gambler's fallacy.
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### Re: Given that you have a brother...

Assume that the population is 50/50, then the fact that your brother is male, detracts from that bias to 50%- for males and 50% for females. This suggest that there is a higher % chance that you are a female.
If the difference is only 1 extra M than F, then it is 50/50.
Any other variation will, by default, place the % chance onto that higher % sex.

The population percentage variation could not be translated down to a percentage of being born M or F. The population is about those who are alive, not those being born. Males may have a higher birth rate, but a correspondingly higher death rate, thus the percentage could swing either side of 50/50 depending on when the point is chosen.

However, taking not population, but birth percentages, then a better determination of whether you are M or F is possible.
If (as Skeptical) alludes to, that there is a 50/50 chance of being born M or F, then there is a greater chance of being female.

However, if Real Life birth rates are taken as the starting point, then whichever sex has the higher % , ensures that you have a higher % of being the same sex as the %.
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### Re: Given that you have a brother...

I toss 10 fair coins.

Is it reasonable to say that there will be exactly 5 heads and 5 tails?

If I hide one of the coins and look at the other 9, does that change the probability that the hidden coin is heads?

Like Skep says, it's the gambler's fallacy. The chance for a specific individual is 50/50, regardless of any other individuals.
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bane2571
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### Re: Given that you have a brother...

skeptical scientist wrote:It's not true that in N people, 50% are male. What is (roughly) true is that each baby, at conception, has a roughly 50% chance of being a male baby. If we are interested in knowing Alex's sex, knowing the sex of people other Alex doesn't give us any new information about the Alex's sex. The idea that it does is essentially the gambler's fallacy.

Mmm, I was looking at it as a pre-existing population with a 50/50 makeup of which me and my brother are members. The question isn't phrased that way so it was really the wrong way of going about it but:

Imagine 50 white balls and 50 black balls, each are randomly handed out to 100 people, if the person next to you has a black ball what is the odds of you having a black ball? That is the way I approached the question.

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### Re: Given that you have a brother...

Go to reality, I've seen more males with brothers than females. females usually have sisters. (i dont put sauce on the steak)

but seriously, there's a 50% 50% chance of that happening. I have realised that being an older one or younger one doesn't affect the probability.

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### Re: Given that you have a brother...

bane2571 wrote:Imagine 50 white balls and 50 black balls, each are randomly handed out to 100 people, if the person next to you has a black ball what is the odds of you having a black ball? That is the way I approached the question.

Yes, in that situation, knowing that they have a black ball decreases (negligibly) the chance that you also have a black ball. But that situation isn't analogous to the situation in the topic, because the situation in the topic doesn't have the number of males and females predetermined before they're distributed... the numbers of males and females are expected to be 50/50 at the end, but it could be on either side of that - it's only 50/50 on average.

Code: Select all

enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}
[he/him/his]

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### Re: Given that you have a brother...

im with the 66% chance that i'm female.
Spoiler:
regardless of whether there are other siblings

we can be

male male
female male
male female

with the older sibling listed first.

so if I knew I had an OLDER brother then it would be a 50% chance i'm either.
male male
male female
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### Re: Given that you have a brother...

You can be with whoever you like, but I'm very sure that the Goldstein got it entirely right. For instance, you have the size of the family fixed, and you also only count "male male" as one person with a brother being male when its two.
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### Re: Given that you have a brother...

50%.
Spoiler:
You are removing two possibilities out of each series.

A) You are the male, the rest of the family is females. (You have no brother in this situation.)

B) You are the female, the rest of the family is females. (Nobody has a brother.)

Seeing as you are removing an equal number of being female, or being male, it remains at 50%.

Examples:

Spoiler:
Family of 2:
MM, FM, MF, FF
We remove MF and FF because you are the Male, and have no brother, or you are the sister, and have no brother.
Leaves MM, FM. 50%

Family of 3:
MMM, MMF, MFM, FMM,
FFM, FMF, MFF, FFF.

Again we remove MFF, and FFF.
Leaving MMM, MMF, MFM and FMM, FFM, FMF. 50%

"Order" here isn't necessarily who came first or who is oldest. If you want to count MF and then count FM as both being possiblities of having a brother because you could be "younger or older" then you have to count it twice against you, because you could also be the "younger or older" female. The same is true with MM (counting twice for) and FF (counting twice against). It's simpler to keep yourself static and count each combination once.

Edit: a bit of clarity.
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Vieto
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### Re: Given that you have a brother...

Spoiler:
50%, not 33% or 66%.

the possible combinations are:

you -- sib
m -- m
m -- f
f -- m
f -- f

leaving a 50% chance of being male.

now, do note that this is true because you have a reference point. If the question was phrased: "There are 2 siblings. One is male. What are the odds that the other sibling is male?" Then the possible combinations would be:

m-m
m-f
f-m
f-f
and the odds are 33% that the other sibling is male.

ttnarg
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### Re: Given that you have a brother...

bane2571 wrote:Mmm, I was looking at it as a pre-existing population with a 50/50 makeup of which me and my brother are members. The question isn't phrased that way so it was really the wrong way of going about it but:

Imagine 50 white balls and 50 black balls, each are randomly handed out to 100 people, if the person next to you has a black ball what is the odds of you having a black ball? That is the way I approached the question.

In a famly with 50 girls and 50 boys, every one has a borther so there is a 50/50 chance the one that is you is male or female.

In a random famly with 100 children there is a 1 in 2^100 chance that your in a famly with all all girls so there are 100 girls who dont have a brother. but there is also 100 chances in 2^100 that your in a famley with 1 boy and 99 girls so there are 100 boys that are not counted. of all the reset that might be you there are equal chance that your a boy or girl.

If you knew that your famley was going to have equal number of boys and girls and you knew that before you where born that you had a brother then the odds of you being a boy would be less. But that is not even close to what we talking about.

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### Re: Given that you have a brother...

Vieto wrote:
Spoiler:
50%, not 33% or 66%.

the possible combinations are:

you -- sib
m -- m
m -- f
f -- m
f -- f

leaving a 50% chance of being male.

now, do note that this is true because you have a reference point. If the question was phrased: "There are 2 siblings. One is male. What are the odds that the other sibling is male?" Then the possible combinations would be:

m-m
m-f
f-m
f-f
and the odds are 33% that the other sibling is male.

alright i read it as "There are 2 siblings. One is male. What are the odds that the other sibling is male?" so i concede to 50-50.
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douglasm
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### Re: Given that you have a brother...

sugarhyped wrote:
Vieto wrote:
Spoiler:
50%, not 33% or 66%.

the possible combinations are:

you -- sib
m -- m
m -- f
f -- m
f -- f

leaving a 50% chance of being male.

now, do note that this is true because you have a reference point. If the question was phrased: "There are 2 siblings. One is male. What are the odds that the other sibling is male?" Then the possible combinations would be:

m-m
m-f
f-m
f-f
and the odds are 33% that the other sibling is male.

alright i read it as "There are 2 siblings. One is male. What are the odds that the other sibling is male?" so i concede to 50-50.

The other way to think of it, without pinning down which one is designated male, is:
Spoiler:
The question/statement is not about pairs (or sets, if you consider larger family sizes) but about individuals. Yes, MM MF and FM all count and twice as many of those have a female as not, but MM has two people with a brother rather than one. The count is two males with brothers from the first pair, one female from the second, and one female from the third. Two of each gender, so an even 50-50 split.

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### Re: Given that you have a brother...

sugarhyped wrote:im with the 66% chance that i'm female.
Spoiler:
regardless of whether there are other siblings

we can be

male male
female male
male female

with the older sibling listed first.

so if I knew I had an OLDER brother then it would be a 50% chance i'm either.
male male
male female

A more accurate table
brother-male you-female
you-female brother-male
you-male brother-male
brother-male you-male

neither older brother nor younger brother change anyfing.

hawkmp4
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### Re: Given that you have a brother...

Me321 wrote:About 51.456310679611650485436893203883% chance that I am male.

http://en.wikipedia.org/wiki/Sex_ratio

You tried to be clever but you missed the assumptions...
I think it's pretty clear that there's a 50/50 chance here.
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bellannaer
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### Re: Given that you have a brother...

It is slightly more likely that you are male.

Spoiler:
There is only one case where the gender of one sibling affects the gender of another, identical twins.

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### Re: Given that you have a brother...

bellannaer wrote:It is slightly more likely that you are male.

Spoiler:
There is only one case where the gender of one sibling affects the gender of another, identical twins.

Not only this, but some people are more inclined to have children of a particular gender, whereas nobody is more inclined to have an equal ration of sons and daughters, so having a brother does make it slightly more likely that you are male in the real world.

However, the problem at hand makes it pretty clear that the gender of every child is randomly determined at birth with an even probability of each gender. It isn't supposed to reflect reality.

Blayze III
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### Re: Given that you have a brother...

Spoiler:
Assuming no parents are trying to skew the results with the previously mentioned birthing practices (If total children = 2, stop having children. If Child1 = Male, have Child 2. If Child1 = Female, stop having children) then there should be no difference in chances of being male or female based upon the fact that you have a brother.

Imagine a family with 10 children. The cases where at least one child doesn't have a brother are as follows, with the gender and number out to the side:

MFFFFFFFFF 1 Male
FMFFFFFFFF 1 Male
FFMFFFFFFF 1 Male
FFFMFFFFFF 1 Male
FFFFMFFFFF 1 Male
FFFFFMFFFF 1 Male
FFFFFFMFFF 1 Male
FFFFFFFMFF 1 Male
FFFFFFFFMF 1 Male
FFFFFFFFFM 1 Male
FFFFFFFFFF 10 Female

Total Males: 10
Total Females: 10[

This hold true for any off-spring size.

N = Number of off-spring parent has
Total Number of Children possibilities = N x 2^N
Half of all possibilities are Male. Total Male possibilities = (N x 2^N)/2
Half of all possibilities are Female. Total Female possibilities = (N x 2^N)/2
Number of Males without brothers = N
Number of Males with brothers = (N x 2^N)/2 - N
Number of Females without brothers = N
Number of Females with brothers = (N x 2^N)/2 - N

Therefore, there is an equal chance of someone with a brother being either Male or Female.

Ralp
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### Re: Given that you have a brother...

Contrast: "Given that your parents have a son, is it more likely that you're male or female?"
Spoiler:
Unless I'm mistaken this is the slightly different question that has involves a probabilty of 2/3 for reasons some people are posting. Given that you have one sibling anyway. I could have worded that first sentence better but I don't care.

Eebster the Great
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### Re: Given that you have a brother...

Ralp wrote:Contrast: "Given that your parents have a son, is it more likely that you're male or female?"
Spoiler:
Unless I'm mistaken this is the slightly different question that has involves a probabilty of 2/3 for reasons some people are posting. Given that you have one sibling anyway. I could have worded that first sentence better but I don't care.

The probability to this question depends on how many kids families have. If all families have an infinite number of kids, the probability is still 50-50. If families have only one kid, the probability is now 100-0. For two-kid families, the probability is 75-25. Et cetera.

phlip
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### Re: Given that you have a brother...

Eebster the Great wrote:For two-kid families, the probability is 75-25.

2/3, I think. The general form is that, in a n-kid family, where the size of the family is independent of the genders of the kids (so the parents aren't doing something like "have kids until we have a son, then stop" or something), the chance of being male given your parents have a son is 2^(n-1)/(2^n-1). So 1/1, 2/3, 4/7, 8/15, ...

Code: Select all

enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}
[he/him/his]

Eebster the Great
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### Re: Given that you have a brother...

phlip wrote:
Eebster the Great wrote:For two-kid families, the probability is 75-25.

2/3, I think. The general form is that, in a n-kid family, where the size of the family is independent of the genders of the kids (so the parents aren't doing something like "have kids until we have a son, then stop" or something), the chance of being male given your parents have a son is 2^(n-1)/(2^n-1). So 1/1, 2/3, 4/7, 8/15, ...

klenwell
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