Which day of the week (Sunday, Monday, etc.) is the probability largest to fall on the 13th of a random month, in a random year?

Or is this probability the same for each day of the week?

(In reply to re: solution and discussion by Elisabeth)

"But this means that, if you have considered the interval 2001-2400, your counting would have been different, and yet other values would arise if you took the 2010-2309 period. So, 400 years are not a significant sample."

Of course the 2010-2309 period would result in different numbers: that is only a 300-year period, while the Gregorian calendar has a 400-year cycle.  However any 400 successive years will have the same distribution of dates vs days of the week in the Gregorian calendar.  Thus if the period were 2010-2409, it would have the same distribution as 2001-2400.  The calendar for 2401 is exactly the same as for 2001, beginning on a Monday and 365 days long, and 2402 has the same calendar as 2002, all the way through to 2409 being the same as 2009. So swapping those sets of 9 years has no net change.

So the distribution from 2001-2400, significantly for counting purposes, is the same as for 2401-2800, and for 2801-3200, etc., the reason again being that (1) the leap year pattern is the same and (2) each of these 400-year periods begins on a Monday.  And the coincidence of Mondays (that is, a particular day of the week, that by our boundary choice is Monday) results from the number of days in 400 years being an exact multiple of seven (365.25 x 400 - 3 = 146097 = 20871 x 7 exactly).

By the way, I took 2000-2399, but that's the same as 2001-2400 because 2000 had the same calendar as 2400 will have.  The 400 year cycle I chose began on a Saturday, Jan. 1,2000, and Jan 1, 2400 will begin on a Saturday also.

Edited on August 30, 2004, 8:25 pm

Posted by Charlie on 2004-08-30 20:19:42