A small course has 3 students, $B$, $C$ and $D$. Based on the previous testscores of these students we know that $B$ usually scores $80$% correct on the exam questions, $C$ scores $60$% and $D$ scores $40$%. The anonymous test that is now corrected has $4$ correct answers out of $8$. What is the conditional probability distribution for who took this particular exam?
This just doesn't want to work. I know Baye's theorem is of use here and possibly the Law of total probability.
Let $B=1, \ C=2$ and $D=3.$ Now also let $A_i, \ i=1,2,3$ be the event that person $A_i$ took the test and $X= \ $number of correctly answered question on the test. We are thus looking for
This did not make anything simpler. The only thing I now in the RHS is that $P(A_i)=1/3.$