Question

quiz has 5 questions. multiple choice with 4 possible answers each. a student randomly guesses on all 5 questions.

1) what is the probability of that student getting a passing grade(60% correct is passing)

2) what is the probability of that student getting a B or better (a low B is 80%)

3) what is the probability of that student failing the quiz?

Answer 1

Given: There are 5 multiple choice questions in a quiz and each question has 4 possible answers. Only one option out of 4 could be correct. Therefore, the probability of guessing a question correctly would be 1/4.

If there are n questions , then the probability of answering x questions correctly, if answers are chosen at random, is given by Binomial distribution as below

P(X=x)= (n C x)*(p)^x *(q)^n−x

here , p=1/4 (one option out of 4 is correct) and q=3/4 (3 option out of 4 are incorrect)

a) The probability of that student getting a passing grade(60% correct is passing)

i.e. P(three questions are correctly answered)

Consider YYYNN as first three right and next two wrong ,

we know that, the probability of YYYNN (first three right, next two wrong) is (1/4)^3 *(3/4)^2

but there are (5C3) total ways for three right and two wrong.

Therefore, we multiply probability (1/4)^3 *(3/4)^2 by (5C3)

P(X=3)= (5 C 3)*(1/4)^3 *(3/4)^2 = 0.087891

b) the probability of that student getting a B or better (a low B is 80%)

i.e. P(4 or more questions are correctly answered)

P(X>=4)= P(X=4)+ P(X =5)

= (5 C 4)*(1/4)^4 *(3/4)^1 + (5 C 5)*(1/4)^5 *(3/4)^0

             =0.014648 + 0.000977

             =0.015625

c) the probability of that student failing the quiz

i.e. P(none of the questions are correctly answered (all questions are wrongly answered))

P(X=0)= (5 C 0)*(1/4)^0 *(3/4)^5

            = 0.237305