# Which days the holidays occur on

Has anyone put together a list of how often each event in the Jewish calendar occurs on a given day of the week? For instance, Yom Kippur happens X% on Shabbat, Y% on Monday, Z% on Wednesday, etc.?

I saw an answer for fast days, but not for anything else.

The molad for each month is evenly distributed across the week over a long enough period of time.

If you look at the rules for the calendar you'll see that:

Rosh Hashanah for a given year is the day its molad falls, unless pushed off by one of the following four rules. 1) Rosh Hashanah would be the first, fourth or sixth day of the week, 2) the molad is 18 hours or more into the day, 3) at the start of a leap year, that number for the third day of the week is 9 and 204/1080 hours into the day, 4) when following a leap year, that number for the second day of the week is 15 and 589/1080 hours.

We can ignore rule two because it doesn't change the distribution at all. If we ignore rules three and four because they are extremely rare and barely affect the distribution, we find that Rosh Hashana is twice as likely to fall on Monday, Thursday or Saturday as it is to fall on Tuesday (because all times it would have been Sundays, Wednesdays or Fridays are pushed off to the the next day).

Or in short: Monday 2/7ths of the time, Tuesday 1/7th of the time, Thursday 2/7ths of the time and Shabbat 2/7ths of the time. (The precise numbers with the other rules included are 28.03%, 11.51%, 31.88%, and 28.57% respectively, but the difference is rarely relevant practically.)

All the other holidays from Purim to Simchat Torah are a fixed number of days from Rosh Hashana so you can easily find the four options for each and which is the "rare" one that is only 1/7th of the time: Purim on Friday, Shushan Purim on Saturday, Pesach starting on Sunday, Shavuot on Monday, 17 Tammuz and Tisha b'Av on Sunday, Yom Kippur on Thurday, Sukkot and Shmini Atzeret on Tuesday.

• The more complicated question is the distribution of the first day of Chanukah - with six possible days to choose from. Aug 18, 2023 at 12:12
• @JoelK I calculated that once, it's in my tur at home. When Im there I'll add it. I never calculated for 10 tevet or tu vshvat Aug 18, 2023 at 12:55
• Correct me if i'm wrong but I think that Purim Pesach and Shavuos are not always a fixed amount of days from RH because Cheshvan and Kislev have variable number of days. Aug 18, 2023 at 15:05
• An analytic solution -- bravo! I would have just relied on someone else's Python calendar code, spat out two centuries of data, and summed it up that way. Thank you for something so elegant! Aug 18, 2023 at 15:34
• @Dave they are a fixed number of days before the next rosh hashana. For our purposes (aggregate statistics) it doesn't matter which direction we go Aug 18, 2023 at 16:11