Let's say a specific exam has 3 levels (I, II, III). The candidates who pass the first exam are then eligible to take the next level of the exam. Let's say the pass rates for levels I, II, and III are .57, .73, and .85 respectively. Supposed 3000 people took level I, 2500 took level II, and 2000 took level III. Supposed one student is selected and he says he passed the exam. What is the probability that he took the level I exam? I tried drawing a tree to list out all the possibilities but I still don't know what the question is asking.
Here's what my tree looks like: https://i.stack.imgur.com/GiGxe.png
Let $E_n$ be the event that the student took the Level $n$ exam. You know how many students took each so assuming there is no bias in selecting the student from among the examinees...
$$~\mathsf P(E_{\rm I})~=\\ \mathsf P(E_{\rm II})~=\\ \mathsf P(E_{\rm III})~=$$
Let $S$ be the event that the student passed the exam. You've been given the values of the conditional probabilies of passing a given exam: $$~\mathsf P(S\mid E_{\rm I})~=~0.57\\ \mathsf P(S\mid E_{\rm II})~=~0.73 \\ \mathsf P(S\mid E_{\rm III})~=~0.85~$$
You are looking to find the conditional probability of taking an exam given that the exam taken was passed. Use the above to calculate this. $$\mathsf P(E_{\rm I}\mid S)~=$$
Fill in the blanks.