What is probability that the 13th day of randomly chosen month is a Friday?

We want the probability that the 13th day of a randomly chosen month is the 13th.

The naive answer is 1/7 as it would appear that each date is equally likely to be one of the seven days of the week.

There are 14 different calendars that we must consider. For each Jan 1, we consider what happens if it is Mon, Tues, etc... for the rest of the year. This creates seven different possible calendars. But we must also consider leap years so that creates seven more.

For non-leap years:

(1) If Jan 1 is a Monday then Jan 13th is a Saturday. Feb 13th is 31 days later and is on a Tuesday. Then Mar 13 is also a Tuesday. Apr 13 is 31 days later and is on Fri. May 13 is 30 days later on a Sunday. June 13 is 31 days later on a Wed. July 13 is 30 days later on a Fri. Aug 13 is 31 days later on a Mon. Sep 13 is 31 days later on a Thur. Oct 13 is 30 days later on a Sat. Nov13 is 31 days later on a Tues. Dec 13 is 30 days later on a Thur.

So that year has 12 months and has 2 Fri 13ths.

(2) If Jan 1 is a Tuesday there are 2 Fri 13ths.
(3) If Jan 1 is a Wednesday there is 1 Fri 13th (in June)
(4) If Jan 1 is on Thursday there are 3 Fri 13ths.
(5) If Jan 1 is on Fri there is 1 Fri 13th (in Aug.)
(6) If Jan 1 is on Sat there is 1 Fri 13th (in May.)
(7) If Jan 1 is on Sunday there are 2 Fri 13ths.

For leap years we have 2,1,2,2,1,1,3

So there are 14(12)=168 months and 24 of them will have a Fri 13th. So the probability of selecting a random month with a Fri 13th is 24/168=1/7