Given that you have a brother...
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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.
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.
Re: Given that you have a brother...
Spoiler:
So, you sacked the cocky khaki Kicky Sack sock plucker?
The second cocky khaki Kicky Sack sock plucker I've sacked since the sixth sitting sheet slitter got sick.
The second cocky khaki Kicky Sack sock plucker I've sacked since the sixth sitting sheet slitter got sick.
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:
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?
Re: Given that you have a brother...
Spoiler:
Spoiler:
Last edited by Goldstein on Tue Apr 13, 2010 10:16 pm UTC, edited 1 time in total.
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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 societywide 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.
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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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%)
Re: Given that you have a brother...
But I don't have a brother.
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.
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.
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:
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:
Spoiler:
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 1/2^{k} 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 1/2^{k1} chance of having k1 children and 1/2^{k1} chance of having k children. So the expected number of children will be $$\frac12+\frac24+\frac38+...+\frac{k1}{2^{k1}}+\frac{k}{2^{k1}}=\sum_{i=1}^\infty \frac{i}{2^i}\sum_{i=k}^\infty \frac{i}{2^i}+\frac{k}{2^{k1}}=2\sum_{i=1}^\infty \frac{i+k1}{2^{i+k1}}+\frac{k}{2^{k1}}$$ $$=2\frac{1}{2^{k1}}\left(\sum_{i=1}^\infty \frac{i}{2^{i}}+\sum_{i=1}^\infty \frac{k1}{2^{i}} \right)+\frac{k}{2^{k1}}=2\frac{1}{2^{k1}}(2+k1)+\frac{k}{2^{k1}}=2\frac{1}{2^{k1}}.$$
Therefore, on average per family in this group, there will be 11/2^{k} boys (with 1/2^{k} brotherless boys) and 11/2^{k} girls (with 1/2 brotherless girls), so among children with brothers, there will be 12/2^{k} boys and 1/21/2^{k} 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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Re: Given that you have a brother...
Spoiler:
Alternate answer:
Spoiler:
Re: Given that you have a brother...
About 51.456310679611650485436893203883% chance that I am male.
http://en.wikipedia.org/wiki/Sex_ratio
http://en.wikipedia.org/wiki/Sex_ratio
Re: Given that you have a brother...
Spoiler:
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 preexisting 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 %.
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.
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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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 preexisting 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.
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.
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Re: Given that you have a brother...
im with the 66% chance that i'm female.
Spoiler:
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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%.
Examples:
"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.
Spoiler:
Examples:
Spoiler:
"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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Re: Given that you have a brother...
Spoiler:
Re: Given that you have a brother...
bane2571 wrote:Mmm, I was looking at it as a preexisting 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:
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 5050.
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Re: Given that you have a brother...
sugarhyped wrote:Vieto wrote:Spoiler:
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 5050.
The other way to think of it, without pinning down which one is designated male, is:
Spoiler:
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Re: Given that you have a brother...
sugarhyped wrote:im with the 66% chance that i'm female.Spoiler:
A more accurate table
brothermale youfemale
youfemale brothermale
youmale brothermale
brothermale youmale
neither older brother nor younger brother change anyfing.
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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Re: Given that you have a brother...
It is slightly more likely that you are male.
Spoiler:
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Re: Given that you have a brother...
bellannaer wrote:It is slightly more likely that you are male.Spoiler:
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.

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Re: Given that you have a brother...
Spoiler:
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:
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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:
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 5050. If families have only one kid, the probability is now 1000. For twokid families, the probability is 7525. Et cetera.
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Re: Given that you have a brother...
Eebster the Great wrote:For twokid families, the probability is 7525.
2/3, I think. The general form is that, in a nkid 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^(n1)/(2^n1). So 1/1, 2/3, 4/7, 8/15, ...
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Re: Given that you have a brother...
phlip wrote:Eebster the Great wrote:For twokid families, the probability is 7525.
2/3, I think. The general form is that, in a nkid 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^(n1)/(2^n1). So 1/1, 2/3, 4/7, 8/15, ...
Yeah, you're right. My bad.
Re: Given that you have a brother...
This problem reminded me of the Monty Hall Paradox, whose Wikipedia page, aptly enough, contained a link to the Boy or Girl Paradox.
The answer as far as I can tell:
The answer as far as I can tell:
Spoiler:
password: m{xkcd.com/this.user}
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