Question

Consider a multiple choice exam of 20 questions. Suppose each question has four choices and only one of them is correct. If a student is going to guess answers, what is the probability that he answers

i.) all questions correctly?

ii.)more than 10 questions correctly?

iii.)more than 5 and less than 10 correctly?

iv.) How many Questions do you expect him to answer correctly?

Answer 1

Here p = 0.25
n = 20

As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

i)
P(X = 20) = 0.25^20

ii)
P(X > 10) = 0.0039

iii)
P(5 < X < 10)
= P(6 <= X <= 9) = (20C6 * 0.25^6 * 0.75^14) + (20C7 * 0.25^7 * 0.75^13) + (20C8 * 0.25^8 * 0.75^12) + (20C9 * 0.25^9 * 0.75^11)
P(6 <= X <= 9) = 0.1686 + 0.1124 + 0.0609 + 0.0271

P(6 <= X <= 9) = 0.369

iv)
expected questions = 0.25 * 20 = 5