Question

a quiz consists of 20 multiple-choice questions. each with 4 choices but only 1 answer, while a student decided to randomly guess all the answers in the quiz.

1. find the probability for passing the quiz when the passing grade is 60%

2. find the probability to get the first correct answer within the first 3 attempts

3. find the expected number of attempts to get the first correct answer

Answer 1

1. P(Correct) = 1/4 = 0.25

Passing grade is 60% so student has to get atleast 20*0.6 i.e. 12 correct.

Hence,

P(12 or more correct)

= 1 - P(11 or less correct)

= 1 - binom.dist(11, 20, 0.25, True) [Excel Formula]

= 0.0009

2. P(First correct within 3 attempts)

= P(Correct in 1st) + P(Wrong in 1st; Correct in 2nd) + P(Wrong in 1st and 2nd; Correct in 3rd)

= 0.60 + 0.40*0.60 + 0.40*0.40*0.60

= 0.936

3. The distribution of variable X which represents the number of questions attempted to get the first correct answer will be geometric distribution. Hence,

Expected number of attempts to get the first correct = 1/p = 1/0.25 = 4