Historically, approximately 1/125 of all births were fraternal twins and 1/300 were identical twins. The probability that Elvis was an identical twin is approximately . . .

In 3000 births, we would expect 3000/300 = 10 sets of identical twins. Roughly half of those we would expect to be boys. That's 5 sets of boy-boy identical twins.

In 3000 births, we would expect 3000/125 = 24 sets of fraternal twins. One fourth would be boy-boy, one-fourth would be girl-girl, one fourth would be boy-girl, and one fourth girl-boy. Therefore six sets would be boy-boy.

So, out of 3000 births, five out of eleven sets of boy-boy twins would be identical. Therefore the chances that Elvis was an identical twin is about 5/11.

Isn't it (1/300) / (1/125 + 1/300) = 5/17 ~ .29?

No, it's 5/11. See http://www.stat.columbia.edu/~gelman/book/ and go to the solutions to exercises in the second edition, Exercise 1.6.

Here's how I think of it:

In 3000 births, we would expect 3000/300 = 10 sets of identical twins. Roughly half of those we would expect to be boys. That's 5 sets of boy-boy identical twins.

In 3000 births, we would expect 3000/125 = 24 sets of fraternal twins. One fourth would be boy-boy, one-fourth would be girl-girl, one fourth would be boy-girl, and one fourth girl-boy. Therefore six sets would be boy-boy.

So, out of 3000 births, five out of eleven sets of boy-boy twins would be identical. Therefore the chances that Elvis was an identical twin is about 5/11.