How common is Shabbos Chol Hamoed?
Calculating the possibilities with both Succos and Pesach
I was recently asked a question about the likelihood of having Shabbos Chol Hamoed. More specifically, how often each of the following scenarios occur in a given year:
1. Shabbos Chol Hamoed Succos only
2. Shabbos Chol Hamoed Pesach only
3. Both
4. Neither
As mentioned before, the months with variable length (Cheshvan, Kislev, and Adar 1 in a leap year) fall out between Rosh Hashanah and Pesach, so if Rosh Hashanah falls out on a certain day of the week it only tells you the date of the preceding Pesach and not the upcoming one. With that in mind, let’s break down when Shabbos Chol Hamoed does and does not happen.
For Succos (which always falls out on the same day of the week as Rosh Hashanah), there is a Shabbos Chol Hamoed when it begins on Monday, Tuesday, or Thursday but not Shabbos. For Pesach, there is a Shabbos Chol Hamoed when it begins on Tuesday or Thursday but not Shabbos or Sunday. Rosh Hashanah is always 2 days later than Pesach, so a Thursday or Shabbos RH means the preceding Pesach had a Shabbos Chol Hamoed while a Monday or Tuesday RH means it did not. In terms of probability, there is about a 5/7 chance of having a Shabbos Chol Hamoed Succos and about a 4/7 chance of having a Shabbos Chol Hamoed Pesach.
Based on what determines each Shabbos Chol Hamoed, we can use Rosh Hashanah on consecutive years to determine each scenario:
1. Monday/Tuesday/Thursday Rosh Hashanah followed by Monday/Tuesday RH the following year
2. Shabbos Rosh Hashanah followed by Thursday/Shabbos RH
3. Monday/Tuesday/Thursday Rosh Hashanah followed by Thursday/Shabbos RH
4. Shabbos Rosh Hashanah followed by Monday/Tuesday RH
Given that there are only a few options of how many days can be in a year, it goes without saying that not every combination is possible. As mentioned before, there are 14 year types (7 each for leap years and non-leap years) so we can determine which ones occur under each scenario:
1. There are 5 year types for this scenario:
a. Monday to Monday leap year – 30 day Cheshvan/Kislev
b. Tuesday to Monday leap year – 29 day Cheshvan, 30 day Kislev
c. Thursday to Monday non-leap year – 29 day Cheshvan, 30 day Kislev
d. Thursday to Tuesday non-leap year – 30 day Cheshvan/Kislev
e. Thursday to Tuesday leap year – 29 day Cheshvan/Kislev
2. There are 3 year types for this scenario:
a. Shabbos to Thursday non-leap year – 30 day Cheshvan/Kislev
b. Shabbos to Thursday leap year – 29 day Cheshvan/Kislev
c. Shabbos to Shabbos leap year – 30 day Cheshvan/Kislev
3. There are 5 year types for this scenario:
a. Monday to Thursday non-leap year – 29 day Cheshvan/Kislev
b. Monday to Shabbos non-leap year – 30 day Cheshvan/Kislev
c. Monday to Shabbos leap year – 29 day Cheshvan/Kislev
d. Tuesday to Shabbos non-leap year – 29 day Cheshvan, 30 day Kislev
e. Thursday to Thursday leap year – 30 day Cheshvan/Kislev
4. There is only 1 year type for this scenario:
a. Shabbos to Tuesday non-leap year – 29 day Cheshvan/Kislev
It should be clear from the above that scenarios 1 and 3 are the most common, with scenario 2 being somewhat less common and scenario 4 being relatively rare. As a (very) rough estimate, scenarios 1 and 3 occur roughly 35% of the time, scenario 2 about 25% of the time, and scenario 4 only about 5% of the time. For reference, scenario 4 is the non-leap year type where Mikeitz doesn’t coincide with Chanukah (which I’ve never discussed before) and the only year that has Mishpatim/Rosh Chodesh/Shekalim. Either way, the numbers should not come as a huge surprise given that it’s more likely to have Shabbos Chol Hamoed on Succos then Pesach yet the latter still happens more than half the time.
The above calculations are obviously for a Hebrew year (as that’s the calendar in question), but I think it’s important to quickly answer the question for a secular year as well. The main reason why it would be valid to ask the question this way is because the Torah lists Pesach first and Succos last when discussing holidays. The opening Mishnah of Masechet Rosh Hashanah qualifies this by calling the first day of Nisan the Rosh Hashanah of holidays. Additionally, it’s much easier to answer the question this way because we only need to look at one Rosh Hashanah.
The same four scenarios in a secular year play out as follows:
1. Rosh Hashanah on Monday or Tuesday
2. Rosh Hashanah on Shabbos
3. Rosh Hashanah on Thursday
4. Never
Here, scenario 1 is the most common at about a 3/7 chance with scenarios 2 and 3 having about a 2/7 chance apiece.
As always, let me know if I got anything wrong!