Gender Probabilities for Twins

Date: 08/27/2006 at 13:17:09
From: Melissa
Subject: Twin gender probability

Hi Doctor Math.  I am pregnant with twins--sex unknown.  Since neither 
is older than the other, what is the probability of having any gender 
combination?  I remember from college genetics that each birth is 
mutually exclusive, therefore the probability of any combination of 
boy or girl is 1/2, i.e. each child will either be a boy or a girl.

My husband is using the bb, gg, bg, gb theory of probability, and 
that would increase the odds of having a boy and a girl to 1/2, while 
having a bb or gg would be 1/4.  My point is that bg and gb are the 
same combination, so his probability isn't correct.  What do you 
think?  I saw your string on a similar question, but it doesn't 
address twin births, and I wanted to be certain that I was thinking 
about this correctly.  Thanks!

Date: 08/27/2006 at 21:13:40
From: Doctor Peterson
Subject: Re: Twin gender probability

Hi, Melissa.

As a twin myself (and my identical twin brother is also a Math
Doctor), I HAVE to take this question!

If there were only fraternal twins, then your husband would be right.
Even with twins, you can distinguish them (firstborn/secondborn,
favorite/nonfavorite, or whatever), so BG and GB are not the same.  As
an example, I was expected to be a girl (because my heartbeat was
weaker, and they didn't have ultrasound yet).  We could be 
distinguished, that is, even without knowing which was a boy and which
was a girl or what our names would turn out to be.  So you could make
up a table of the four equally likely cases:

               Dave
           | boy | girl|
       ----+-----+-----+
       boy | 1/4 | 1/4 |
  Rick ----+-----+-----+
       girl| 1/4 | 1/4 |
       ----+-----+-----+

That gives probablities of 1/4 (two boys), 1/4 (two girls), 1/2 (one
of each).

Now in reality, you also have to bring in the probability that the
twins turn out to be identical--in which case they MUST both be the 
same sex.  (I understand there are other odd possibilities, but I'll
neglect those.)

Suppose the probability of having fraternal twins is F, and of having
identical twins is I.  Then, given that you have twins, the 
probabilities of their being identical is I/(F+I); we end up with this
table:

               BB     BG     GB     GG
              
               I                    I
  Identical  ------               ------
             2(F+I)               2(F+I)

               F      F      F      F
  Fraternal  ------ ------ ------ ------
             4(F+I) 4(F+I) 4(F+I) 4(F+I)

So the probability of two boys, or of two girls, is

  2I + F
  ------
  4(F+I)

and the probability of one of each is

     F
  ------
  2(F+I)

A couple sites I looked at said that F = 1/125 and I = 1/300.  Another
site said that 1/3 of all twins are identical.  Others give different
numbers.  If I take that first pair of numbers, we get the probability
of two boys, or of two girls, is 0.3235, and the probability of one of
each is 0.3529.  So each probability is about 1/3.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/