An unprepared student randomly guesses the answers to all of the 20

Question

An unprepared student randomly guesses the answers to all of the 20

multiple choice questions in an exam. If each question has four possible

choices, what is the probability that the student scores at least 70 % on

the exam?

[Hint: Answering a multiple choice question here is a Bernoulli trial with

probability 0.25. There are 20 independent Bernoulli trials and the number

of correct answers is a Binomial random variable.]

Answer 1

We want to calculate the probability of making at least a 14 on a 20-question test.

 

P(20) = 20!/(20!0!)(1/4)²⁰(3/4)⁰

 

P(19) = 20!/(19!1!)(1/4)¹⁹(3/4)

 

P(18) = 20!/(18!2!)(1/4)¹⁸(3/4)²

 

P(17) = 20!/(17!3!)(1/4)¹⁷(3/4)³

 

P(16) = 20!/(16!4!)(1/4)¹⁶(3/4)⁴

 

P(15) = 20!/(15!5!)(1/4)¹⁵(3/4)⁵

 

P(14) = 20!/(14!6!)(1/4)¹⁴(3/4)⁶

 

 

P(14 or more) = P(20) + P(19) + ... + P(14) = 0.0000295

 

The student should have studied.