# Probabilities in a multiple choice test

70 students take a multiple choice test with 100 questions. Each question has 4 possible answers— a, b, c, d — and there is only one correct answer per question. Student 1 gets 22 wrong answers and Student 2 gets 19 wrong answers. What is the probability that Students 1 and 2 have 13 wrong answers that are the same (eg. both picked “c” when c is one of the three possible wrong answers to a question) out of the 100 questions?

Does the fact that Student 1 had 13 out of 22 wrong answers that were the same as Student 2’s wrong answers prove that the two students were cheating off each other’s tests?

• This does not have an answer unless you assume that the students are using random guessing. – Michael R. Chernick Jun 21 '19 at 17:40
• Pretty high if there's "hot tip" going around that in this class you should pick (c) when in doubt because on the last few exams there were lots of correct (c) answers. Not unlikely that two students would believe it. – BruceET Jun 21 '19 at 18:25

$$p(i matches) = \left( \frac{1}{3} \right)^i \cdot \left( \frac{2}{3} \right) ^ {19-i} \cdot {{19}\choose{i}}$$ $$\sum_{i=13}^{19}{p(i)} \approx 0.0019$$