I was sitting over dinner with the mispaha (family) talking about the calendar and I said that Addar 2 happens every seven years, but my dad said that it happens every four. So, my question is: how often is there an Addar II?
The extra Adar happens when there is a leap year -- we add a month to preserve the lunar properties of the calendar. Seven in nineteen years are leap years, so every 2-3 years on average.
From Judaism 101:
Adar I is added in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle.
(Technically it is Adar I (aka Adar Alef) that's the additional month, and Adar II (Adar Bet) is the last month of every year. But in a year with just one it's just Adar.)
I am assuming that your question refers to the current "fixed" calendar, so it seems that you have numerous answers on that.
I wanted to add that historically, this was not always the situation during the time that the Sanhedrin existed and prior to that. I just completed a class on the history and the workings of the Judaic calendar, which you will find via a link on my blog. The 1st or 2nd entry has a link to a Google shared drive, and you can view the PowerPoint file in the "Session 2" folder.
In summary - The Sanhedrin's main concern was to ensure that 16 Nissan (2nd day of Passover) occurs after the Spring equinox. If not, an extra month was added.
While that was the main criteria, Sanhedrin could also declare a leap year if the barley crop had not yet ripened, if the fruit trees did not bloom, or if the roads were muddy and / or bridges were washed out because of heavy rains during the rainy season. There are a few other reasons, as well, which I mention in the blog. I gleaned most of my materials from Maimonides Hilchot Kiddush Hachodesh.
In the fixed calendar it happens 7 times in 19 years.
The lunar month, according to our tradition (we announce the "molad") in fact is exactly 29 days, 12 hours, 44 minutes and one "chalak", an 18th of a minute, or 42524 minutes + 1 chalak, or 765433 chalakim. (Hey, almost an easy number to remember, isn't that).
If we ignore "leap seconds" and we know that the Gregorian calendar is 365.2425 days which is 525949.2 minutes or 9467085.6 chalakim.
Divide these numbers and we get 12.36827469, the actual number of months in a year. Now I, as a mathematician, know what a recurring fraction is, and that give good approximations.
The first few are: 12 25/2 37/3 99/8 136/11 235/19
when you get to this point the next factor is 18. A high factor means you have a good approximation.
Our calendar does actually drift forward a bit. (Remember we usually still follow the Julian calendar, e.g. when we start saying V'Tein Tal Umatar outside Israel and when we recite Birchat Hachama). However our calender doesn't drift forward to the extent the Julian calendar does.
The next value is 4366/353 which means if we really wanted to stop the drift then, after 18 cycles of 19 years, we should insert a cycle of 11 years, then resume as before.
Using the Julian calendar our target would be 12.36852866 and 235/19 is (as I mentioned) slightly shorter so we'd have to add in more extra months. Actually 25 of lots of 19 then a run of 8, giving us 5974 months in 483 years.