Question

A student takes random guesses at each problem on a 15-problem quiz. Each problem is a multiple choice question with 5 possible solutions. What is the probability that the student gets a B (80.0% - 89.5%) on the quiz?

Answer 1

Solution:

We are given

Number of problems in a quiz = n = 15

Probability of correct answer = p = 1/5 = 0.20

q = 1 – p = 1 – 0.20 = 0.80

80% of 15 = 15*0.80 = 12

89.5% of 15 = 15*0.895 = 13.425 ≈ 13

np = 15*0.20 = 3

nq = 15*0.80 = 12

np < 5, nq > 5, so we can’t use normal approximation.

So, we have to use binomial distribution for finding the required probability.

We have to find the probability that the student gets a B (80.0% - 89.5%) on the quiz.

That is, we have to find P(12≤X≤13)

P(12≤X≤13) = P(X=12) + P(X=13)

P(X=x) = nCx*p^x*q^(n – x)

P(X=12) = 15C12*0.20^12*0.80^(15 – 12)

P(X=12) = 15C12*0.20^12*0.80^3

P(X=12) = 455*0.20^12*0.80^3

P(X=12) = 0.00000095420416

P(X=13) = 15C13*0.20^13*0.80^2

P(X=13) = 105*0.20^13*0.80^2

P(X=13) = 0.0000000550502

P(12≤X≤13) = P(X=12) + P(X=13)

P(12≤X≤13) = 0.00000095420416 + 0.0000000550502

P(12≤X≤13) = 0.0000010092544

Required probability = 0.0000010092544