One of the first things to do with claims of "likely" is to determine the scope. The choice of timescale determines whether this claim is as valid as stated.
The Gregorian (and Julian, for that matter) calendar has 14 possible single year calendars. Pick a weekday for January 1st and whether it is a leap year, and that gives the calendar (that's how those perpetual calendars in almanacs work).
In a normal, stable, situation those calendars repeat in a cycle of 28 years. Within that normal cycle of years, the distribution of 13ths to Fridays is exactly 1/7 of the total. The seven leap year calendars have a total of 12 Fridays which fall on the 13th of the month. The total for the 7 non-leap year calendars is the same.
In the 28 years, you get each leap year once and each non-leap year three times,
for a total of 48 Fridays on the 13th of the month.
Since 28 years have 336 months, and 48 * 7 is 336, this is exactly even.
(Pedants may look here for more detail)
The discussion above focused on the case of normal year cycles within the Gregorian calendar. That means that it holds for timescales which do not cross exceptional years (that includes my lifespan and the lifespans of everyone likely to be reading this).
But there are times when the normal cycling of years is disrupted. In a sense, that was the whole point of establishing the Gregorian calendar (but that is a discussion for a different page).
Within the Gregorian calendar, centennial years (those evenly divisible by 100) are only leap years if they are also divisible by 400. For example: 1800, 1900 and 2100 are not leap years. (1700 wouldn't have been, had the calendar change been adopted by then.)
When a timeframe crosses one of these, the frequency of the calendar repetitions gets skewed, and the distribution is no longer even. This is because replacing that single leap year with a non leap year not only replaces that year. It also disrupts the cycle of years such that an entire section of the sequence is omitted.
Any particular year may have one, two or three 13ths falling on Friday. If the section dropped has an average number of Friday the 13ths greater than 12/7 (1.7142) then the overall average will drop. If the omitted section has an average less than 12/7, the overall average will rise.
Crunching the numbers for a 400 year segment yields this distribution:
The expected number for each day with an even distribution would be
400 years * 12 months / 7 calendar days = 685.714
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Copyright 1997, Drew Lawson.
[Last updated: 18 April 1997]