What is the chance of a Friday being the 13th of the month?

The leap year rules for the Gregorian calendar state that every 4th year
is a leap year, except that every 100th year is not a leap year, except
that every 400th year is a leap year.

How many days are there in 400 years?  400*365 + 100 - 4 + 1 = 146,097
[Editor: was 146,907], which just happens to be exactly 20,871 weeks.
This means that the calendar repeats itself exactly with a period of
400 years or 4,800 months.  This means that the average can be taken
over 400 years instead of over an infinite time span.  In a span of
4,800 consecutive months:

        Number of Sunday-the-13ths:    687
        Number of Monday-the-13ths:    685
        Number of Tuesday-the-13ths:   685
        Number of Wednesday-the-13ths: 687
        Number of Thursday-the-13ths:  684
        Number of Friday-the-13ths:    688
        Number of Saturday-the-13ths:  684

The probability of the 13th of the month falling on Friday is
        688 / 4,800 = .14333...,
which is slightly more than 1/7 = .142857... .

The probability of a Friday being a Friday-the-13th is
        688 / 20,871 = .032964400364141631929471515... .

-- Dave Dodson <dodson@convex.COM>