Assuming that it is equally like to be born in any of the 12 months.
Reasoning for 1/2: Probability that both are born in the month of Jan is 1/12*1/12 as event must happen one after another. This probability is same for every other month. The event can happen in any of the 12 months, we need to add 1/24, 12 times. So we got 1/2.
Reasoning for 1/12: There are 144 total outcomes and 12 favorable outcome. All the event are equally likely so the probability is 1/12.
Where am I doing wrong? Please suggest.