Amat,
You are actually right. I made a typo when I mentioned that two people share the same birthday. Instead, I meant birth month. So I was effectively addressing the same question that you attempted to answer. But there can be two interpretations to the Pa's question:
1) what is the possibility that at least two people share the same birthday?
2) what is the possibility that only two people share the same birthday?
However, the answer to the second question is not 13/24 but almost .0054.
Too simplify things, the probability of ONLY two people sharing a birth month in a room of 13 is almost zero
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