Question

4. The score of a student on a certain exam is going to be a number between 0 and 1. (Think of it as the percent of correct answers on the exam.) Suppose the student passes the exam if their score S is at least 0.55. If the probability density function of a students score S is given by: (4x0sxs} f(x) = {4-4x sxsi 10 elsewhere. (a) What is the probability that the student fails the exam? (b) If the student passes the exam, what is the probability that their score S was at least 0.75. (Hint: This is a conditional probability question.)

Answer 1

a) Probability that the student fails the exam is computed here as:

70.5 p0.55 P(S < 0.55) = f(x)dx + f(x)d.o J0,5

70.5 0.55 P(S < 0.55) = (4.xdx + | (4 – 4.c)da J0.5

P(S <0.55) = 2(0.54) + (4(0.55 -0.5) – 2* (0.552 - 0.5))

P(S < 0.55) = 0.5+ (0.2 - 0.105) = 0.595

Therefore 0.595 is the required probability that the student fails the exam here.

b) Given that a student passes the exam, probability that their score was at least 0.75 is computed using bayes theorem as:

PS > 0.75|S > 0.55) = PS > 0.75) 1- P(S < 0.55)

P(S > 0.75|S > 0.55) = 1 S0.75(4 - 4.c) de 1 - 0,595

P(S > 0.75 5 > 0.55) - (4(1 – 0.75) - 2(14 -0.752) 1 -0.595

P(S > 0.75|S > 0.55) = 1- 0.875 1-0.595 = 0.3086

Therefore 0.3086 is the required probability here.