Let (F, M, C) be a triple where each of F, M and C is a number from 1 to 365 (birthdays of mother, father, and child, respectively). Then the total number of such triples is 3653. The number of triples where each number corresponds to the 7th of some month is 12 * 11 * 10. Namely, there are 12 variants for mother; for each of those there are 11 variants for father; and for each of the first two there are 10 variants for child. If every triple (F, M, C) is equally likely, then the probability is the ratio .
Note that this probability corresponds to a given day of the month (7th or any other fixed date <= 28). The probability that all three have birthdays on the same day of the month in different months without specifying which day it is will be of course greater (by about a factor of 30).