# Twin Probabilities Problem

1 in 8 sets of twins are sets of identical twins. While talking to Jessica, you learn that she has a twin named Emily. What's the probability that Emily and Jessica are identical twins?

Approximately

2/9

because

non-identical twins are about equally likely to be of opposite sex as of same sex, whereas identical twins are always of the same sex. So you get an extra 1 bit of evidence for "identical" over "non-identical" when you learn that Jessica's twin is called Emily.

Why

2/9 rather than 1/4? Well, imagine that pairs of twins come in groups of 32; in each group 4 pairs are identical, two MM and two FF. The other 28 pairs are non-identical: 7 MM, 7 MF, 7 FM, 7 FF. (The first one in each pair is the one you talk to.) A priori you're equally likely to encounter any of these, but learning that they're both female reduces the options to 7 FF non-identical pairs and two FF identical pairs. Thus, 2/9.

Reality is a bit more complicated because

sometimes people have strange names (maybe Emily is a boy) or change gender (maybe Emily has XY chromosomes but now identifies as female), and for all I know there may be funny effects where identical twins are more often of one sex than the other, or where fraternal twins are more likely to be of concordant sex (imagine some condition that kills babies in the womb, but affects boys and girls differently...). But presumably that 1/8 figure is only an approximation anyway, and these complications probably don't make a large difference.

Assuming that Jessica and Emily are both girls, the probability that we have identical twins with the same gender is:

$P_1 = \frac18$

The probability that we have fraternal twins with the same gender is:

$P_2 = \frac78 \times \frac12 = \frac{7}{16}$

The probability twins with the same gender are identical is therefore:

$\frac{P_1}{P_1 + P_2} = \frac29$