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?

Edit: We can calculate the probability if we assume that each question has a 60% likelihood of being correctly answered by either student.
For each question, the odds that the students will both choose the correct answer is $.60 \times .60=.36$. In this case, their answers will coincide. The odds that one student chooses the correct answer and one doesn't is $2 \times .60 \times .40 = .48$. In this case, their answers will not coincide. The odds that neither will choose the correct answer is $.40 \times .40 = 0.16$, in which case (assuming their choice is random) the odds that they will have the same guess is $1/3 = .333$. So, the odds per question that the guesses will coincide is $.36\times1 + .40\times0 +.16*.333 = .413$. 
Now we want to know the odds that they guess the same for 25 questions; this is $.413^{25} = 2.5 \times 10^{-10}$.
