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This is an interesting problem, with a couple interpretations. I'll go through a couple different ways of looking at it, and answers.

The first is: given that a family has one son and one daughter, what is the probability that the daughter was born on the same day as the father and the son was born on the same day as the mother.

Well, assuming that there are no seasonal biases in the conception process, the son has a 1/365 chance of matching the mother, and the daughter has a 1/365 chance of matching the father for a total chance of 1/133225, or about 7.5 in a million.

The second question could be: in a family with two children, what is the chance that one is a son with the same birthday as the mother, and the second in a daughter with the same birthday as the father? This is slightly different, because in the first question the two children were specified to be a boy and a girl, and in this question all we know is that their are two children. They could be boy-boy, boy-girl, girl-boy, and girl-girl, so there is only a 2/4 = 1/2 chance that the children will be boy-girl. Therefore the probability becomes 1/266450, or about 3.7 in a million.

The final question could be as in your subject line: in a family with two children, what is the probability that both children match both parents. This needs to be broken down into two parts: if the mother has the same birthday as the father (which happens with probability 1/365), there is a 1/365/365 = 1/133225 chance of both children also matching the same birthday. If the mother and father have different birthdays (364/365) then there is a 2/365 chance for the first child to match, and a 1/365 chance for the second child to match the remaining parent. Altogether this gives a 729/48627125 or 15 out of a million chance.

No matter how you view it, an extremely unlikely event!

Of course for simplicity I ignored the effects of leap years, and twins!

All the best,
Mark Huber