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Who have more sisters: boys or girls?

A simple question with some surprising twists.

[As featured on Radio 4's Today programme, 18th January 2018]

Here's a puzzle to ponder when you are next with your family.

In this country, on average, who have more sisters - boys or girls?

To keep it simple, let's assume that boys and girls are evenly split, and the sex of the next child born is entirely random. In other words, having children is just like flipping a coin.

If both of those assumptions are true, then what is the answer?   Do girls typically have more sisters, or boys, or is it the same for both? Think about it for a moment, because there's a spoiler - and some surprises - coming.

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When I first heard this puzzle, my intutition was the same as many other people's.  A girl can't be her own sister, so in mixed families, the boys all score more sister points than the girls do.  This suggests that boys will therefore, on average, end up having more sisters than girls do.

But this is forgetting families who have children all of the same sex.  In these families, boys have no sisters while girls (in all-girl families) have at least one.

Imagine two-child families.  There are four possible combinations.  The children can come along in the order BB, BG, GB or GG, and each of these four combinations is equally likely.  The four boys here have a total of two sisters between them.  The four girls also have a total of two sisters.  In other words, in two child families boys and girls have an average of 0.5 sisters.  In three child families, the possible combinations are BBB  BBG  BGB GBB BGG GBG GGB and GGG.  Here, the 12 boys have a total of 12 sisters, and so do the girls, so boys and girls have an average of one sister..  And this pattern of boys and girls having the same number of sisters continues for higher numbers of siblings.

So the answer to the simple puzzle is that on average boys and girls have the same number of sisters, (Though we don't know what the average number of sisters per person in the UK is.  My guess is that on average we all maybe have about half a sister each.)

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But I made a couple of huge assumptions at the start.  What about the real world?  

I believe there is statistical evidence that the sex of the youngest child in a family is not entirely independent of the sex of their older siblings.  Once you've had two girls, the chance of your third child being a girl is very slightly higher than 50%.  (I'd love to see the actual stats on this.)  If it's true, then this means that GGG/BBB families will be more common than other combinations, and the implication is that on average girls will end up with slightly more sisters than boys.

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And in any case, there is a real world situation that definitely affects the answer.  Twins.  There have always been twins, but since the introduction of IVF, the number of twins (and triplets etc) has increased significantly.  Twins are usually the same sex, and hence they mean girls will on average have more sisters than boys, and boys will on average have more brothers.  My thanks to Alan Leigh and Mike Cornish who both alerted me to this after the sisters puzzle appeared on the Today programme.

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But there are more twists, Girls and boys aren't exactly split 50/50.  Historically, slightly more boys have been born than girls.  Does this make a difference?  Remarkably, it doesn't.  If boys and girls are born with a probability of, say, 55% and 45% respectively, then although this will affect the total number of girls, the average number of sisters will be the same for boys and girls (if we ignore twins).   It gets more complicated, however, when you allow for the fact that girls live longer than boys.  This means that as a family ages, it will tend to become more dominated by girls. Does this change the average number of sisters for boys relative to girls? My first hunch was yes...then no...and now I'm not sure.

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There's another real world factor.  Parental decisions on whether to have another child are sometimes influenced by what children they've had already.  This can work in two ways.  Suppose parents have had two boys so far.  I've known parents who said at this point: "we'd love to have a girl, let's try for another baby".  But I've also known parents who said: "we have two boys, we can't risk having a third so let's stop now".  I have no idea which response is more common, but (I think) it's going to distort the distribution of sisters in some way.

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And then there's China.  Until recently China had a single child policy.  But there was a little bit of leeway: if you had a son you had to stop, but if you had a daughter, you were allowed to try for a second child.  This meant that in China, if people obeyed the law then the only family combinations that were possible were:  G   B  GB and GG.  Sisters only cropped up in GB and GG families, and since these two outcomes were equally likely, if we assume Chinese families always try to have a boy, then girls would have, on average, 2/3 of a sister each, while boys would have 1/3 of a sister.  So - in theory - in China girls average more sisters than boys.  And since China has such a big population, maybe this tips the world average in favour of girls having more sisters.

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My thanks to Hugh Hunt who first raised the question with me, and to John Haigh with whom I've discussed some of the results. Maybe there are more twists to come.