On Monday and Thursday, we say a longer Tachanun. On a fast day (where the weekday prayers are still said), Selichot are said in addition to Tachanun.
It feels like there are many more fasts that fall out on Monday and Thursday. Is this statistically true, or does it just feel that way because I always remember having to say the longer Tachanun in addition to the Selichot? (availability bias)
Answer
Considering Monica's point about year distributions, here's a refinement of Gershon's data (using the frequency table on Remy Landau's Hebrew Calendar page, here):
Tzom Gedaliah falls on Monday or Thursday in the year types גכה, זחא, זשג, גכז, זחג, זשה. This is 40.08% of all years.
Asarah B'Teves can't occur on Monday, but it can be on Thursday, only in the year type הכז. This happens to be the single most common keviah, occurring in 18.05% of all years.
Taanis Esther can be on Monday or Thursday quite often (largely because of the cases where it is moved up from Shabbos) - indeed, the only time it's not is when Purim itself is on Thursday, which is in the year types הכז, בשז, גכז. The fast is therefore on Monday or Thursday 71.97% of the time.
Shiva Asar B'Tammuz can't fall on Monday, but it can be on Thursday, in the year types בשה, גכה, בחה, זשה. These are 28.57% of all years.
Same thing for Tisha B'Av: it's on Thursday in 28.57% of all years.
So the only one that occurs on Monday or Thursday a majority of the time is Taanis Esther, with Tzom Gedaliah being a distant second.
All told, if you consider the full Hebrew calendar cycle of 689472 years, with five fasts in each one, then you'll have that a fast occurs on Monday or Thursday 37.45% of the time.
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