Conditional probability in birth

Question

Suppose that there are two endangered species of polar bear. The only difference between the species is in the probability of giving birth to twins. Species X gives birth to twins 10% of the time, otherwise giving birth to a single cub. Species Y gives birth to twins 20% of the time, otherwise giving birth to a single cub. Suppose that you are working for a captive breeding program. You have a new female bear of unknown species. She just gave birth to twins. What is the probability that her next birth will also be twins?

Answer 1

There is not enough information to answer this question. It depends on the relative proportions of Species X and Species Y. If almost all polar bears are Species X, the the probability that the next birth will be twins will be close to $10\%$, but if almost all polar bears are Species Y, then the probability that the next birth will be twins will be close to $20\%$.

If we assume that the relative frequencies of species X and Y are $p$ and $1-p$, respectively, then we can answer the question in terms of $p$. Indeed, in that case, the probability that the bear is Species X is equal to $\frac{.1p}{.1p+.2(1-p)}=\frac{p}{2-p}$, and the probability that the bear is Species Y is equal to $\frac{2-2p}{2-p}$. Then, the probability that the next birth will be twins is equal to $.1$ times the first of those probabilities, plus $.2$ times the second one, which comes out to $$\frac{4-3p}{20-10p}.$$