I think the math is bit off. Let's say you have the 1,2,3, and 4th of the month off. Assuming total randomness in the days off (unlikely, as csomar commented), your friend has 1/30 chance of having the 1st off, 1/30 of having the 2nd, ... so in total 4/30 chance of having a day off at the same time as you do in a given month.
1/30 chance of having the first day off? They have four days too, it should be 4/30 each of the four days, for overall odds of just over half. I ran a quick monte carlo and it said the same thing.
Oops. I didn't know why I squared 4/30, which is better since it is once every ~7.5 months.
Its not as non-random as csomar suggested, since you could have the day off any day of the week. However you usually get a day off once a week. I guess it should be better estimated at 1/7 chance of having the same day of, per week. That gives roughly 1 day every 2 months
