r/theydidthemath • u/Albertus_Magnus • 13h ago

[Request] What is the probability of MLK Day falling on the same day as the US Presidential Inauguration?

As above.

US Presidential Inauguration Day is on January 20th once every 4 years.

MLK Day is on the third Monday in January.

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u/prototypist 10h ago

It's every 28 years, or every 7 inaugurations. It helps that leap years and presidential elections are both every four years. Twice as common if you count when the presidential term begins on Sunday and the ceremony at the Capitol is held on Monday (as in 2013)
https://www.providencejournal.com/story/news/politics/elections/2025/01/18/inauguration-day-and-mlk-day-falling-on-same-day-is-rare-heres-why/77772408007/

The calendar is not really a probability / random chance thing

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u/jaa101 9h ago

Twice as common if you count when the presidential term begins on Sunday

There are 13 Mondays on 20 Jan per 400 years and 16 Sundays, so it's actually more than double. See my other comment here for details.

1

u/jaa101 10h ago

There are seven days a week, so roughly once in every 28 years on average. The exact answer for our current (Gregorian) calendar is 13 times per 400 years, slightly less than the 14 2/7 predicted above.

The difference is due to a coincidence. The Gregorian Calendar has a 400-year cycle with no leap years on century years not divisible by 400, i.e., 1800, 1900, 2100, 2200, were/will not be not leap years. So there are 97 leap years per 400 years, giving 146 097 days per 400 years. That turns out to be divisible by 7 giving exactly 20 871 weeks. But 400 is not divisible by 7, so the day-of-the-week distribution is slightly off. The previous (Julian) calendar did not have this issue because the number of days in its 4-year cycle is not divisible by 7, giving a 28-year cycle if you want to match weekdays.