Question

A lazy professor gave two tests A and B to a class of n students and assigned marks Ai and Bi, respectively, to student i, for i = 1,2,...,n, uniformly and independently at random with values from the set of possible grades 1,2,...,k. What is the probability that some student receives the same grade in both tests?

Answer 1

We can think as A and B are two discrete random variable which can take any value between 1 to k. And for each random variables there are n observations.So the pmf of both random variable would be

P(A=a)=1/k where a=1,2,...,k P(B=b)=1/k where b=1,2,.....,k

= 0 otherwise = 0 otherwise

So, the joint pmf of A and B is P(A=a,B=b)= 1 / k² where a= 1,...,k b=1,...,k

= 0 otherwise

Now we have to find for i^th student P(a_i=b_i)

So, P(a_i=b_i) = P(a_i=t,b_i=t)= 1 / k²

Hence required probability is 1/k²