How often does a month have 5 Saturdays and 4 Sundays?

Answer 1

When a month has 28 days, this can't happen.

When it has 29 days (February of every leap year) it happens only when the first day of the month is a Saturday.

When a month has 30 days, then if the first day of the month was a Saturday, you would also get 5 Sundays. To get 5 Saturdays and only 4 Sundays, then, means the first day of a 30-day month has to be a Friday.

For a 31-day month, you need the first day of the month to be a Thursday, so that days 3, 10, 17, 24 and 31 are Saturdays and days 4, 11, 18 and 25 are Sundays.

A rough calculation would suggest that the probability of getting the Saturday on the first day of a 29-day month is 1 in 7, that of getting it on the second day of a 30-day month is 1 in 7 and getting it on the third day of a 31-day month is 1 in 7.

So any given month (except February on non-leap years) has a chance of about 1 in 7. 28-day months happen 97 out of every 4800 months in the current Gregorian calendar. So for the other 4703 months out of 400 years, there's a 1 in 7 chance. According to my calculations, it's a 14.00% chance of it happening in the long run, or about once every 7 months on average.