If a 20 question multiple choice test (with 4 choices a-d) can be modeled as a Binomial random variable, answer the following questions concerning the test:

Question

(a) What is p for any question if you are guessing randomly?

(b) What are the mean (µ) and variance (σ 2 ) for this distribution?

(c) What is the formula for this particular distribution?

(d) What is the probability of answering exactly 15 questions correctly if you are guessing randomly on each question?

(e) What is the probability of answering 3 or fewer questions correctly if you are guessing randomly?

Answer 1

a) p = 1/4 = 0.25

b) mean = np = 20 * 0.25 = 5

variance = np(1-p) = 20 * 0.25 * 0.75 = 3.75

c) 20!

----- 0.25^x 0.75^(20-x) where x is number of correct answers

x!(20-x)!

d) substitute for 15 for x in above equation and compute probability

e) substitute 0, 1, 2, 3 for x in above equation, compute and sum probabilities