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?

Simply assuming the average score is not enough. For example, take a look at the following test:


*

*Question $1$: What is $1+1$?

*Question $2$: Prove conclusively that $P=NP$.


This test will have an average score of $50\%$, and the number of students that answer the same on all questions will be equal to the number of students taking the test.
On the other hand, you can ask a multichoice question about differential equations to a set of $6$ year old children. In that case, you can more or less assume that the answers are randomly distributed.
