Question

There are 100 multiple-choice question on exam, each having responses a, b, c, d. Suppose that a student has a 70% chance to answer each question correctly, and all the answers are independent. If the student needs to answer at least 40 questions to pass, what is the probability that the student passes? Use normal approximation.

Answer 1

There are 100 multiple-choice question on exam, each having responses a, b, c, d. Suppose that a student has a 70% chance to answer each question correctly, and all the answers are independent. The student needs to answer at least 40 questions to pass. If we consider each question to be a bernoulli trial, where either he answers a question correctly or he doesn't.

let us define X_i to be the result on the i^th question, then , dependending on whether he answers the question correctly or not. It is given that the student has 70% chance of correctly answering a question, so for all i,

Then, the total of questions he answers correctly will be given by

Since, he answers each question independently the sum of all the independent bernoulli distribution will result in a binomial distribution. X ~ Bin(100,0.7), since p> 0.5 and np=100*0.7 > 5, n(1-p)=100*0.3>5, so we can use normal approximation to binomial probabilities.

Probability that the student passes =

since, , when n is very very large.

It seems reasonable that he has sure chance of passing, because he has very high probability of correctly solving each mcq.