Question

A recent college graduate is planning to take the first three actuarial certification exams over the course of the next year, the first one in June, second in July and third in August. If she fails any, she will not take the remaining exams. The probability she passes the first is 0.9. Given that she passes the first exam, she has a 0.75 chance of passing the second and given that she passes both the first and second, she has a 0.65 chance of passing the third. (a) What is the probability she passes all three exams? (b) Given that she did not pass all three exams, what is the probability that she failed the second? (c) Given that she did not pass all three exams, what is the probability that she failed the third?

Answer 1

a) Probability that she passes all three exams is computed using the multiplication rule of probability as: = 0.9*0.75*0.65 = 0.43875

Therefore 0.43875 is the required probability here.

b) Probability that she fails the second exam is computed here as:

= Probability that she passed the first * Probability that she failed the second given that she passed the first = 0.9*(1 - 0.75) = 0.225

Also probability that she did not pass all three exams is computed as:

= 1 - Probability that she passed all three exams = 1 - 0.43875 = 0.56125

Therefore the required probability now is computed using the Bayes law as:

= 0.225 / 0.56125

= 0.4009

Therefore 0.4009 is the required probability here.

c) Again similar to the above way, probability that she failed the third exam is computed as:

= 0.9*0.75*(1 - 0.65) = 0.23625

Similar to the above part, the probability here is computed as:

= 0.23625 / 0.56125

= 0.4209

Therefore 0.4209 is the required probability her.e