Two students, Andy and Betsy, are preparing for a difficult exam in Random Signa

Question

Two students, Andy and Betsy, are preparing for a difficult exam in Random Signals & Systems. To maximize their learning they decide to study together. Andy and Betsy alternate between the roles of 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 probability 0.10 when he is the teacher. Assume that the event that a very difficult problem gets solved is independent of all other problems. What is the probability that a very difficult problem gets solved? A very difficult problem was solved. What is the probability that it was solved by Betsy? In order to improve their chances of passing the course, Betsy has reluctantly agreed to play the teacher more frequently than Andy. Suppose Andy and Betsy want the probability of solving a very difficult problem to be 0.30. What should be the probability that Betsy plays the teacher?

Explanation / Answer

Probabilty of either of both becoming teacher = 0.5

Probability that andy answering when she is teacher =0.1

Probability that besty answering when she is teacher = 0.4

a) Probabilty that very difficult quest gets solved

= probability of andy solving + probability of besty solving it

= 0.5*0.1 + 0.5*0.4

= 0.05 + 0.20

= 0.25

b) Difficult problem is solved

probability that it is solved by besty

= 0.20 / 0.25

= 0.8

c) Now, probability of besty becoming teacher be p

andy be q

0.3 = q*0.1+p*0.4

3 = q+4p

Also , p+q=1

3 = 1-p+4p

2 = 3p

p = 0.667

hence besty role should be 0.667

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