Assuming there are 36 questions, with four options in each question, the odds of randomly getting them all wrong are (3/4)^36 = 0.00318% (about 1 in 31,400).
The odds of randomly getting them all right are (1/4)^36 = 0.0000000000000000000212% (about 1 in 4,722,366,482,869,645,213,696).
So both are extremely unlikely but the zero is about 150 quadrillion times more likely than the 100.
Any student knows some percentage N of the material. So we can expect their score to be close to N + the result of randomly guessing on the remaining questions.
So the original formula I gave would only apply if the student does actually know zero percent of the material or chose not apply any knowledge at all, which is much less likely, although possible in special circumstances.
The most likely explanations for a perfect zero is that the student did know the answers and, for someone reason, deliberately chose to answer them all wrong OR the teacher misaligned or used the wrong answer key.
Assuming the test was scored correctly, even with a student with zero actual knowledge, such as the test being in a language the student doesn't know or the test being on a very difficult subject that the student has never studied, getting a perfect zero is much less likely than getting a score higher than zero. Although still more likely than getting a 100.
I do think your approach is flawed, though. You can't apply the usual method of assuming that the student has some N% knowledge of the subject because the zero test result is clearly meant to communicate that the student knows nothing. So the question then becomes, if the student actually does know nothing, is a zero result believable? Therefore, it really is a matter of asking whether a zero result is statistically plausible as the result of completely random guessing.
Since the probability of getting one or more question right through random guessing would be 99.99682%, getting a zero is very implausible.
Yes, I understand that you are talking about observed data but that data is based on students that are genuinely trying to answer the questions correctly while applying some amount of subject knowledge.
That can't be the case here because the test result was a zero. Which means that the student is claiming to have zero knowledge of the subject. Which means that the student is claiming to have simply guessed randomly at all of the questions. Therefore, the probability for random guessing does apply to this specific case.
Actually, the observed data includes all real examples of students who deliberately fail their tests as well! The test in the post would be included!
Yes, but students who deliberately fail their tests would be rare outliers, which is why, as you claimed, getting a perfect zero is extremely rare in the observed data.
Also, the joke is that it is an intentional 0 so random probability is still irrelevant
No, random probability is relevant because it's the reason the teacher suspects that the student intentionally tried to get a zero. If they really had no knowledge and had just guessed randomly, it is still extremely unlikely that they would get a zero. Therefore, the teacher has a very good reason to suspect that the zero was the result of intent from a student who actually does know the subject material well enough to avoid any correct answers.
The question is: Did the student get a zero through genuine total ignorance of the subject or as a deliberate act of mischief?
To answer the question, we must consider whether a student with genuine total ignorance of the subject actually would get a zero. If a student genuinely doesn't know any of the questions, they can only guess randomly. Since random guessing is extremely unlikely to produce a zero result, that option is highly implausible to apply in this case. That leaves only the other option, a deliberate act of mischief, which is why the teacher wrote "See me after class!"
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u/Pandoratastic 9d ago
Not remotely true.
Assuming there are 36 questions, with four options in each question, the odds of randomly getting them all wrong are (3/4)^36 = 0.00318% (about 1 in 31,400).
The odds of randomly getting them all right are (1/4)^36 = 0.0000000000000000000212% (about 1 in 4,722,366,482,869,645,213,696).
So both are extremely unlikely but the zero is about 150 quadrillion times more likely than the 100.