>>212253
Hello, and welcome back to our occasional recurring thread, Azum&th!
Your question is a discrete version of a problem on the random distribution of chords on a circle -- there are elegant solutions which get in to calculus to explain, and I'd have to sit down with a book on probability to make sure I was doing it right.
However, with just 6 choices of 12 possibilities, it's possible to crunch the numbers by brute force, simply by generating all 12^6 possibilities and checking whether all 6 choices of birth month lie within 4 months of each other. If we also allow them to be even more compressed (so we might have a nine-month stretch without birthdays, for instance), the probability is 1.35%. If we insist that some two of the birthdays are exactly in some month and month+3, the probability is 1.09%.