the probability for these questions

Question

There two types of students at StudyU: Strong students (S) and exceptional students (E).

In physics 200, 85% of students are strong students (S) and 15% of students are exceptional students (E). An exceptional student has a 90% probability of getting an “A” on each physics 200 exam, and a strong student has a 50% probability of getting an “A” on each physics 200 exam.

1. If a student gets an “A” on the first physics, what is the probability that they are an exceptional student?
2. If a student gets an “A” on the first physics, what is the probability that they are a strong student?
3. In physics 200, assume that each student’s performance on each exam is independent of their performance on each of the other exams. If a student gets an “A” on the first two exams in physics 200: What is the probability that they are an exceptional student?
4. In physics 200, assume that each student’s performance on each exam is independent of their performance on each of the other exams. If a student gets an “A” on the first two exams in physics 200: What is the probability that they are a strong student?

Answer 1

P(E|A) = P(EA)/P(A) =

P(EA) /(P(EA)+P(SA)) =

P(A|E)P(E)/(P(A|E)P(E) + P(A|S)P(S) ) =

.9(.15)/(.9(.15) + .5(.85)) =

.2411

P(S|A) = 1 - P(E|A) = 1 - .2411 = .7589

P(E|AA) = P(EAA)/P(AA) =

P(EAA) /(P(EAA+P(SAA)) =

P(AA|E)P(E)/(P(AA|E)P(E) + P(AA|S)P(S) ) =

.9(.9)(.15)/(.9(.9)(.15)+.5(.5)(.85)) = .3638

P(S|AA) = 1 - P(E|AA) = 1 - .3638 = .6362