Now to the numbers:
1. Assuming that impregnation is equally probable on any given month; and assuming normal variations in gestation time, one can reasonably assume that any given birthday on a given year is equally likely. (P=1/365.25)
2. Assume that impregnation is equally likely in any given year over the ~20 child-bearing years of the mother.
3. So the probability of having a birthday on any given day over that 20 years is 1/(365.25*20) = 1/7305.
4. Assuming that the births are statistically independent (perhaps not a good assumption), the probability of ANY four births occurring on specified dates is (1/7305)4 = ~1/315.
Note again that any specified combination of 4 birthdays is equally likely (or unlikely).
Thank you very much for answering. 1/3000000000000000 sounds like a big ol' number to me! (But it probably isn't in the grand scheme of things...)
Regards,