Possibility that two students answer the same on a 25 question multiple choice test I am trying to get some approximation of the possibility that two students would answer $25$ questions in a row exactly the same.  There are 4 possible responses for each question.  I calculate the random possibility as $4^{25}=1.13\times10^{15}$. Since, however, I assume some amount of familiarity with the material, I divided that by $4$.  I imagine, however, that this is not a precise method.  If I assume an average score of $60%$, is there a more precise model to determine the probability?

While you can come up with a mathematical answer. It would only show how unlikely the possibility of the two students getting the same answers assuming they were to randomly guess. In reality, it is not as unlikely as you would think. For example some questions might have a really deceptive choice and both students pick that choice. Their was a trig test in my Algebra 2 Trig class where two students both failed the test with a score of a 43 and had the same exact answers as each other. Does this prove they cheated? No, they both had their calculators in radians instead of degrees and answered every question correctly. Yes, while mathmatically you can calculate the probability of two students getting the same answers with an average score of 60, but I would advise against it. Just because they have the same answers doesn't mean they cheated...If thats what you are trying to prove.
