I too think answer should be 1/365.
Since we know that Father already has a birthday before kid has a choice of it, probability of father's bday is 1 (365/365). And hence, kid's birthday should be 1/365. But, I am not sure. If the events - father's bday and kid's bday - take place at the same time, then the answer would be 1/365^2. Otherwise, it should be 1/365.
trublu,
the probability of rolling the dice twice and getting the same number is 1/6 and not 1/36.