Question

Nigel has just been given a twenty question multiple choice quiz where there are four possible answers to each question. Unfortunately for Nigel, he has not studied and consequently must guess the answer to each question. a. What is the probability that he will answer at least 70% of the questions correctly?

Answer 1

Here, X is random variable which represents number of questions which are correctly answered out of n=20 questions and there are four possible choice for each question.

X~Binomial(n=20, p=1/4=0.25)

We have to find  probability that he will answer at least 70%(20*70/100=14) of the questions correctly i.e. P(X=14).

Consider,

P(X=x)=ⁿC_x*p^x*(1-p)^n-x

Hence, P(X=14) =²⁰C₁₄*0.25¹⁴*0.75^20-14= 2.569872e-05

Thus,  probability that Nigel will answer at least 70% of the questions correctly is 2.569872e-05.