A student needs to take and pass three exams to pass a course. If she fails an exam she cannot take the subsequent one(s) and fails the course. The probabilities of passing the exams are 0.9, 0.75 and 0.6 consecutively. Given that the student did not pass the course, what is the probability that she failed the th exam, = 1,2,3?
I tried to solve it with conditioning on failing for all i values. But seems like I am missing something.
Thanks for your help.
Fail first exam $p_1=0.1$, fail second exam $p_2=0.9\times 0.25=0.225$, fail third exam $p_3=0.9\times 0.75 \times 0.4=0.27$. Total $=0.595$ As you must see, failure on a later exam results only if previous exams passed.