my sister has just got pregnant with twins, she already has a boy. My
brother in law believes that the probability of having a girl is 50%.
I believe it is higher because they already have a boy. My arguement
is based on the fact that before my nephew was born, there was a 50%
chance that if they had 3 children, they would have a boy. his
arguement is that the caclualtion of the probability has now gone back
to zero as my nephew being born is a historic event. where can I find
a mathmateical arguement to back myself

You won't, because your brother is right. There are various factors
that determine the baby's sex, so it is not exactly like throwing a
dice, but it is still closer to that (in probablity) than to what you
suggest: even if you've already had "3" in one throw, it wouldn't
prevent you from having "3" in the next (or one of te next) throw.

For the record, there are many families where all the siblings have the same sex.

The probability of the baby's gender depends on when it is conceived,
so the percentage from last conception would not come in to play; you
would need to recalculate the probability again: here is an article on
the subject http://www.amazingpregnancy.com/pregnancy-articles/74.html

Also, Here's a form that will predict the baby's gender:
http://www.boyorgirl.com/predict3.htm

Politicalguru-ga,
You are absolutely right on your last remark:  I knew a girl whose
brother was the seventh son of a seventh son  - not that there weren't
some daughters in between (my good luck).
Although many native English speakers don't know it, "dice" is the
plural of "die" in your context (Würfel), whereas "dies" is the the
plural when referring to a die in any other meaning.
Myoarin

According to Bayesian stats, it will depend on whether or not the events :-

Event 1 - Having a boy
Event 2 - Having second and subsequent child

are statistically independent or not. 

If the events are statistically independent - then your brother is right.

If the events are statistically independent, then you are right.

Regards

Kaps

oops spotted a typo ...

According to Bayesian stats, it will depend on whether or not the events :-

Event 1 - Having a boy
Event 2 - Having second and subsequent child

are statistically independent or not. 

If the events are statistically independent - then your brother is right.

If the events are NOT statistically independent, then you are right.

Regards

Kaps

From the CIA Factbook
Sex ratio at birth at birth: 1.06 male(s)/female
so the probability of having a boy is 51.4563%
This ratio is based on world statistics. In USA the sex
ratio at birth is 1.05 male(s)/female so the probability
of having a boy is 51.5122%
statistics about the world
http://www.cia.gov/cia/publications/factbook/geos/xx.html
statistics about the USA
http://www.cia.gov/cia/publications/factbook/geos/us.html

Consider this:

There are three possibilities for the twins: (as noted above the fact
that they already have a child makes no difference)

2 girls
2 boys or
1 boy, 1 girl

Assuming that the chance of having a boy is roughly the same as having
a girl, there are 2 out of 3 possibilities of having a boy. Therefore
the odds are 2/3.
Or about 67%

There is a bit of a problem with these kinds of answers:

While in the population, the probability of having a boy or a girl is
roughly 50/50 (a little higher for boys, a little lower for girls);
that in no way implies that a particular person(or couple)has the same
likelihood.

With no past history, the best guess would be 50% or so.  But there is
history here.  There are some women (or couples)who have a real
propensity to have one or the other.  That propensity is only revealed
by having kids.  Having a girl first says something about your
propensity to have girls.  It doesn't say much, but it does say
something.  Babies are not pennies, and the biological building blocks
are not tossed in the air to mingle together freely.  There can be
real reasons why a couple would have lots of boys or lots of girls,
not just random chance.

Having a girl first changes the expected likelihood of having more
girls only very slightly.  For most people, having a girl reveals
nothing real about a physical predisposition.  But since some people
do have differnences in real propensities, the expected likelihood has
to move.  If I recall correctly, having a girl first raises the
probabilty of having another girl by half a percent or so.  Having two
girls in a row is stronger evidence of a predisposition, so it moves
it around 3% I believe.

None of that says a woman who has had 3, 4, or 5 in a row of anything
cannot have the other.  Random chance cannot be eliminated as being
responsible for an unusual chain.  Nor does a real physical
predisposition mean the other sex is impossible.  I am friendly with a
family that had a girl after seven boys.  The chances that they have a
predisiposition toward boys are high, but not complete. Random or
real, they have their girl.

Also, with twins you have alot more to consider.  Just because there
are three possible outcomes, it doesn't follow that they have the same
probability.

Identical twins will be of the same sex, fraternal twins can be the
same, or opposite.  There are five possible outcomes (BB Identical, GG
Identical, BB Fraternal, GG Fraternal, and BG Fraternal) If identical
twins are more or less common than fraternal twins, you have to take
that into account.

What is true in average is not necessarily true in each case, but only
experience tells you if you have a different likelyhood.  Consider
this:  you are going to be far more likely to have another set of
twins after this than would a woman who had not yet given birth, but
less likely than a woman who had twins the first time.

There ya go.
dhouse