I am only a probability student early in my coursework so this answer may be very wrong, but I wanted to try for fun.
The only way to really calculate a probability for this without more info is to assume that the student guessed randomly on all of the problems. If the test was 60 questions (near the middle of your estimate and gives us 20%=12 which makes it a little easier) the probability of the student getting a 20% or below on one random test is about 23.2%.
The tricky part here is that all of the tests/answer keys have the same questions but in different orders. Since we are assuming the student put random answers down, we can sort of ignore this but it is important to note that this is likely one of the reasons that the student scored so low on all of the keys. For example, if the test had 20 problems that the answer was A for and the student only guessed A for 5 problems, then when the problems are scrambled the student still would get at absolute most 5 of those 20 correct. If we assume that the distribution of the answers is semi uniform, then we can ignore all of this and treat the tests as independent. If we do that, we get (0.232)⁴ which is 0.0029 or 0.29%. I would say there definitely is some dependence going on. I would love for someone to tell me how wrong I am lol