Example 3. A preliminary test is customarily given to the students at the beginning of a certain course. The following data are accumulated after several years:
a) 95% of the students pass the course, 5% fail.
96% of the students who pass the course also passed the preliminary test.
25% of the students who fail the course passed the preliminary test.
What is the probability that a student who has failed the preliminary test will pass the course?
Let A be the event "fails preliminary test" and B be the event "passes course". The probability we want is then P(B|A). So we need P(A∩B) and P(A). P(A∩B) is the probability that the student both fails the preliminary test and passes the course; this is P(A∩B) = (0.95)(0.04) = 0.038. 95% of the students passed the course and of these, 4% had failed the preliminary test. We also want P(A), the probability that students fail the preliminary test. Thus, P(A) is the sum of the probabilities of the two events: "passes course after failing test" and "fails course after failing test". Then,
P(A) = (0.95)(0.04) + (0.05)(0.75) = 0.0755.
(Of the 95% of students who passed the course, 4% failed the preliminary test; of the 5% of the students who failed the course, 75% failed the preliminary test since we are given that 25% passed.)
P(B|A) = 0.038 / 0.0755 = 0.509.
That is, half of the students who fail the preliminary test succeed in passing the course. We are interested in event B ("passes course") given event A. Thus, instead of the original sample space, we consider a smaller sample space. We then want to know what part of this sample space corresponds to event B ("passes course"). This fraction is P(A∩B) / P(A), which is computed.