In a Biology class 85% of students pass the midterm exam. If a student passes the midterm exam, there is a 75% they pass the final. If a student does not pass the midterm exam, there is a 55% chance of passing the final exam.

If a student passes the final, what is the probability that they passed the midterm exam?

Answer:

p(mid exam / final exam)

= 0.8854

Step-by-step explanation

Let 'A' = pass the mid term exam

and 'B' = pass the final exam

given that

p(A) = 0.85

p(B / A) = 0.75

p(B / A^c) = 0.55

p(A / B) = ?

p(A and B) = p(B / A) * p(A) = 0.75 * 0.85 = 0.6375

p(B / A^c) = p(B and A^c) / p(A^c)

=> 0.55 = p(B) - p(A and B) / 1 - p(A)

=> 0.55 = p(B) - 0.6375 / 1 - 0.85

=> 0.55 = p(B) - 0.6375 / 0.15

=> 0.55 * 0.15 = p(B) - 0.6375

=> 0.0825 = p(B) - 0.6375

=> p(B) = 0.6375 + 0.0825

=> p(B) = 0.72

Now

p(A / B) = p(A and B) / p(B)

= 0.6375 / 0.72

= 0.8854