Question

A class has 40 students.

Thirty students are prepared for the exam, • Ten students are unprepared.

The professor writes an exam with 10 questions, some are hard and some are easy.

• 7 questions are easy. Based on past experience, the professor knows that:

– Prepared students have a 90% chance of answering easy questions correctly

– Unprepared students have a 50% chance of answering easy questions correctly.

• 3 questions are hard. Based on past experience, the professor knows that:

– Prepared students have a 50% chance of answering hard questions correctly

– Unprepared students have a 10% chance of answering hard questions correctly

• Each student’s performance on each question is independent of their performance on other questions.

(a) Find the probability that a prepared student answers all 10 questions correctly.

(b) What is the probability that at least one of the 30 prepared students answers all 10 questions correctly. Assume that each student’s score is independent of every other student.

(c) Let P be the number of questions answered correctly by a randomly chosen prepared student, and let U be the number answered correctly by a randomly chosen unprepared student. Find E[P] and E[U]

Answer 1

(a) For prepared students

P(correct I easy) =0.90

P(correct I hard ) = 0.50

P(easy) =7/10=0.70  

P(hard) = 3/10 =0.30

P(correct) = P(correct I easy).P(easy) + P(correct I hard ).P(hard)

= 0.90 *0.70 + 0.50* 0.30

= 0.78

Probability of answering one answer correctly = 0.78

Probability of answering 10 answer correctly = 0.78¹⁰ = 0.0834

(b) Using Binomial probability law

Let X be the number of students answer 10 questions correctly

X follow binomial with n= 30 , p =0.0834

= 0.0735

= 1- 0.0735

= 0.9267

Probability that at least one of the 30 students answer 10 questions correctly = 0.9267

(c)Prepared student

Probability of one answer correct = 0.78

Out of 10 questions , expected number of correct answer , E(P) = 10*0.78 =7.8

Unprepared students

Probability of one answer correct = P(correct I easy).P(easy) + P(correct I hard ).P(hard)

= 0.50 *0.70 + 0.10* 0.30

= 0.38

Out of 10 questions , expected number of correct answer , E(U) = 10*0.38 =3.8