Probability of identical twins, given that they are both boys

Question

Overall, 25% of twins are identical. If we assume both genders are equally likely, what is the probability that her twins are identical? (twins of different genders cannot be identical)

Answer 1

Let $B$ be the event "both boys" and let $I$ be the event "identical." We want to find $\Pr(I|B)$. By the definition of conditional probability, we have $$\Pr(I|B)=\frac{\Pr(I\cap B)}{\Pr(B)}.$$ If we make the usual independence assumptions, we have $\Pr(I\cap B)=\frac{1}{8}$.

To find the probability of $B$, note that $B$ can happen in two ways: (i) identical and both boys or (ii) not identical and both boys. The probability of (ii) is $\frac{3}{4}\cdot \frac{1}{4}$, if we assume that for non-identicals, the sexes of the older and younger are independent.

Thus $\Pr(B)=\frac{1}{8}+\frac{3}{16}$.

Now we have all the ingredients.

Remark: We can do a more informal computation. Imagine the births of $256$ pairs of twins. We will get a pair of identical twin boys about $32$ times. We will get a pair of boys about $32+48$ times. Divide.

Or else we can draw a tree diagram. There is a pair of branches from the "root," labelled Identical and NotIdentical, decorated with the probabilities $0.25$ and $0.75$. Then the Identical branch gives rise to two branches, Boys and Girls, each decorated with probability $0.5$. The NotIdentical branch gives rise to two branches, labelled Boys, Mixed, Girls, decorated with probabilities $0.25$, $0.5$, $0.25$. Now use the tree to find the probability of Identical and Boys, and to find the probability of Boys. Divide.