probability that a year has 53 mondays

Question

We have the years from 2001, 2002, 2003,... to 2010. Say, a year is chosen at random from the listed years. What is the probability that the chosen year has 53 Mondays ?

Answer 1

In $400$ years (a Gregorian calendar cycle) there are $365\times 303+366\times 97 =146097$ days which is $\frac{146097}{7}=20871$ weeks and so there are $20871$ Mondays.

Since $20871=329\times 52+71\times 53$, there are $71$ years with $53$ Mondays, and so the probability that a year has $53$ Mondays is $\frac{71}{400}=0.1775$.

Similar calculations over a $400$ year cycle can show that the $13$th of a month is more likely to be a Friday than another other particular day.

Answer 2

A year with 53 Mondays must start on a Monday, or be a leap year starting on a Sunday, on average a given year having 53 Mondays occurs about once every five or six years (five times in a period of twenty-eight years).

Since 2000 and going up to 2050, the following years have 53 Mondays:

2001 2007 2012 (LY) 2018 2024 (LY) 2029 2035 2040 (LY) 2046

LY = Leap Year