Two students Andy and Betsy are preparing for a difficult ex

Two students, Andy and Betsy, are preparing for a difficult exam in Random Signals & Systems. To etsy alternate between the roles o teacher and pupil, so that at any time, the probability that Andy is taking the role of teacher is 0.50. Betsy, who has the higher GPA of the two students, can solve a very difficult problem with probability 0.40 when she is the teacher. Andy, on the other hand, can solve a very difficult problem with proba bility 0.10 when he is the teacher Assume that the event that a very difficult problem gets solved is independent of all other problems.

Solution

Since it is not not given explicitly I am going to take an assumption that when Andy/Betsy are Pupil they cannot solve the problem.

What is the probability that a difficult problem gets solved?

P(problem gets solved) = P(andy solves the problem) +P(betsy solves the problem)

P(problem gets solved) = 0.5*0.1+ 0.5*0.4 = 0.25

A very difficult problem was solved. What is the probability that it was solved by Betsy?

P(Problem solved by betsy | Difficult problem was solved) = (0.5*0.4)/0.25 = 0.80

What should be the probability that Betsy plays the teacher?

Let P denote the probability the betsy plays the teacher.

P(difficult problem is solved) = P(andy solves the problem)+P(betsy solves the problem)

0.3 = (1-P)0.1+P*0.4

P= 0.2/0.3 = 0.67