Number of Siblings
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
Tenyearold Bert Sampson lives in the town of Springvale, where, out of all households:
2% include five children
7% include four children
14% include three children
31% include two children
16% include one child
30% do not include any children
What are the chances that Bert lives with two sisters and no brothers?
(Consider all children living in any given house as siblings).
Answer
6.646%
Since Bert is a child, he cannot possibly live in a household with no children. Therefore there is a (14*3)/(16+31*2+14*3+7*4+2*5), or 26.582% chance that he lives with two other children (i.e. in a household with three children).
There is a 50% chance that each of these other children is a girl, so there is a 25% chance that they are both girls.
Therefore, the total chance that Bert lives with two sisters is 25% of 26.582%, or 6.646%.
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Comments
Kiroho
Oct 23, 2007
 I don't know where you got 14 out of 70.
According to the problem you set up, we have to assume that he lives in a house with 3 children, otherwise we don't have enough information to do all the probabilities.
Fourteen percent of the homes have 3 children. I'll agree with the 50% chance for each other sibling for a 25% chance of 2 girls. 25% of 14 gives us a 3.5% chance.
Because you didn't specify "exactly" 2 sister or "at least" 2 sisters, we can't calculate for 4 children or 5+ children homes. 
Kiroho
Oct 23, 2007
 I just reread the question. You're right and I was making assumptions that were not correct. I read the question to state that he DID have 2 siblings, but that wasn't stated.
Sorry. 
luckylefty
Oct 24, 2007
 huh???? 
tsimkin
Oct 26, 2007
 There is a relatively big problem with this. I agree that you have to adjust the probability of him being in the family of 3, but he is more likely than you stated. If 14% of the families have three kids, then the adjustment isn't 14/70, but (14*3)/(16+31*2+14*3+7*4+2*5); you are more likely to get a kid from a multikid family. Think instead of a town where there are 2 families; one with 1 child, and the other with 2. If I take a child at random, there is a 2/3 chance that they come from the 2child family. Therefore, the correct answer to this teaser is 6.646%, not 5%. 
Xpurrt
Oct 26, 2007
 loved the bart simpson springfield idea 
Dormousesr
Oct 31, 2007
 I love the teaser, but I don't agree with the answer.
I think the correct answer is 9/79, or 11.392%.
I agree with tsimkin's ajustment of 14%  that it is not 14/70 but 42/158. But I think the chance that the both Bert's siblings are girls is not 25%, but 3/7.
Look, nobody told that Bert is the oldest, so there are 7 possible options (from oldest to youngest): boygirlgirl (BGG), BBG, BGB, BBB, GBG, GBB and GGB  and three of them  1 boy and 2 girls.
Therefore, the chance is 3/7 of 42/158, or 11.392. 
leftclick
Oct 31, 2007
 Firstly, thanks everyone for your feedback. I'm glad it has made everyone think
I do agree that my original answer is wrong (it was approved by at least 7 reviewers... so I wasn't the only one lol).
What tsimkin said was totally correct. You do have to consider the weighting of the number of children per family, like he said. (Another example is a town where half the families have no children and the other half have 5 children  there would be a 100% chance that Bert would have 4 siblings)
This also raises an issue with the part of the teaser that says "2% of families have 5 or more children". To make it possible to calculate the answer, the "or more" part needs to be removed, otherwise, we can never know how many children there actually are. (Imagine if one family had 100 children for example).
Kiroho is correct in saying that I didn't specify "exactly" or "at least" 2 sisters, but I did mean "exactly 2 sisters and no brothers"  I just maybe wasn't explicit enough about that.
Dormouser is also correct in saying that the chances of him having 2 sisters isn't 25%, but I think with the wrong result. She listed 7 of the 8 possibilities for a 3child family, the one she missed is the GGG option, I think because Bert is a boy and therefore can't be in a GGG family, but it is a mistake to simply discount these families as well. There is actually a 3/8 (not 3/7) chance that any 3child family will consist of 1 boy and 2 girls.
To summarise: Of any given 100 families, there are 79 boys (half the 158 total children). 3/8 of the 14, 3child families (=5.25) include exactly one boy and two girls (i.e. there are 5.25 boys per 100 families that have 2 sisters only). Therefore, the answer is 5.25 / 79 = approximately 6.646%, the same answer tsimkin got.
I will put in a correction... 
leftclick
Oct 31, 2007
 OK, somebody beat me to it... I hope it was somebody with the right answer 
Dormousesr
Nov 01, 2007
 I thought about this teaser  and found the way to complicate the solution even more than ever!
Now I do not agree with everybody  lol.
I think the previous ajustment was wrong.
The first thing we have  a boy living in a town.
So we should exclude as impossible variants not only nochild families, but all noboy families (i.e. families either with no children or with only girls  because Bert by no means can live in them).
Noboy families among 1child families amount to 1/2; in 2child families they are 1/4; in 3child  1/8 and so on, therefore we have:
(14*3*7/ / (16/2 +31*2*3/4 +14*3*7/8+7*4*15/16 +2*5*31/32)=28.894%
I do insist that in order to get the final result we should multiply 28.894 by 3/7  not 3/8 (exactly because Bert can not live in GGGfamily)  so the chance is 12.383%. 
Dormousesr
Nov 01, 2007
 I do not know why it happened but instead of in my previous post must stand "8 )". 
leftclick
Nov 01, 2007
 Dormouser, I agree that with your method you do need to multiply by 3/7, not 3/8, but that is because you already multiplied by 7/8 in your numerator (the part that became the smiley), and 3/7*7/8 = 3/8
However you have not accounted for the fact that in multiple child families, there are more than one possible "positions" for Bert. So in a BBG family, this actually counts as 2 families when determining which of the Bs represents Bert, and likewise he has 3 times as much chance of being in a BBB family than a BGG family. This would significantly increase your denominator  by a factor of about 12.383/6.646
I definitely think the answer of 6.646% is correct 
leftclick
Nov 01, 2007
 ...the new improved answer that is 
tsimkin
Nov 05, 2007
 Dormouser and LeftClick  you don't have to make any adjustments to the 25% piece of the calculation, because once you take as a given that Bert has two siblings, order doesn't matter at all. With two siblings, the probability that they are both girls is indeed 1/4. The equallyprobably possible combinations (that could include Bert) are:
Bert GG
Bert GB
Bert BG
Bert BB
G Bert G
G Bert B
B Bert G
B Bert B
GG Bert
GB Bert
BG Bert
BB Bert
You can see that 1/4 (or really 3/12) of these have 2 sisters for Bert. 
Dormousesr
Nov 07, 2007
 Yes, tsimkin, you are absolutely right about 25%. I realized it when leftclick said that there were two "positions" for Bert in BBG families and three  in BBB families.
But I still believe that the rest part of the formula I've written is right... 
Dormousesr
Nov 07, 2007
 So, I think the correct answer is
(14*3*7/8*1/4) / (16/2 + 31*2*3/4 + 14*3*7/8 + 7*4*15/16 + 2*5*31/32) = 7.224% approx. 
Dormousesr
Nov 09, 2007
 I confess  I was wrong, and tsimkin is right  the correct answer is 6.646%. I am so sorry and so ashamed
And thank you for this great teaser! 
reidc80
Jan 08, 2008
 no one seems to have taken into account that more male children are born than females therefore it wouldnt be a 25% chance that both siblings are girls.you would have to factor in the fact that there are likely more male children in the town. 
leftclick
Jan 08, 2008
 reidc, yes there is a slightly different chance of being born male vs. female, but since it doesn't say anywhere what the ratio is (and this varies quite significantly between different countries and periods in history), I think 50/50 is a pretty good assumption to make 
reidc80
Jan 09, 2008
 I agree with you it is an almost insignificant amount but for curiosity I looked it up.In the US assuming the time period is now there are 3%more male children under 18 in households than female children. my source:The anne casey foundation kids count us. 
javaguru
Dec 09, 2008
 Wow, all these comments and answers and none of them is correct. The answer is 9.2563...%.
This is based on the fact that the question does not state exactly two sisters, so the .26582 * .25 = .066456 probability that Bert lives in a house with three childen and two sisters needs to have the .17722 * .125 = .022152 and .06329 * .0625 = .003956 probabilities that he lives in a house with 4 children and three sisters and with 5 children and four sisters.
This all assumes a 50% probability for male/female for each of Bert's siblings. In reality, in addition to the fact that boys are more likely than girls, there is the fact that having had a boy, it is even more likely that his parents had another boy instead of a girl. (The sex of a child is influenced by the father's genes, making the likelyhood that the second child of the father's is the same sex as the first child greater than the probability that the first child was that sex.) 
javaguru
Dec 09, 2008
 By the way, I arrived at the answer above before reading the given answer or any of these comments. The question is still written so as to allow for more than two sisters, so this is the answer to the question as written. 
Johnny
Apr 08, 2009
 Isn't it 14%? If Bert lives with 2 sisters, and no brothers, that means there are 3 siblings in the family. You specificly said the percentage of families with 3 children is 24%. 
leftclick
May 20, 2009
 Wow, I'd forgotten about this teaser as I haven't been on this site for a while. I've looked at it again and I agree with the answer as given (as corrected by tsimkin)
@javaguru, The "exactly" is implied. Perhaps it should be explicit. Do you agree with the answer, assuming that the question actually means "exactly two sisters and no brothers"?
@reidc, As I said, you are right, but I think it is natural to assume a 50/50 split. However maybe it should also be explicit in the question  "In Springvale, exactly 50% of children are girls"? 
Juanito
Jun 25, 2010
 Great teaser.
I see that javaguru keeps trolling abou t.. 
silverhand
Jul 11, 2011
 Sorry, I'm a bit confused about how/why the prob of him coming from a 3 child family isn't simply 14/70. If there are 100 families and 14 have 3 children and you remove the 30 that have no children... then there is a 14 out of 70 chance of a 3 child family. It would be different if 14% of the children came from 3 child families but that isn't the question. Please help. Thanks. 
Castlerock
Feb 26, 2013
 20% of the families with children have three children, and assuming he lives in a family with only 3 childern, which the puzzle suggest, he has a 5% chance 
Dezaxa
Dec 28, 2014
 As with many problems of this kind, the answer is underdetermined. We need to be told how Bert came to be selected. Did someone enter the town and pick a child at random? Did someone pick a family at random and then a child at random from that family? Was it part of the selection process that only a boy could be chosen? Without these, and probably other things, being specified, there is no single correct answer. 
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