A multiple-choice standard test contains total of 25 questions, each with four answers. Assume that a student just guesses on each question and all questions are answered independently. (a) What is the probability that the student answers more than 20 questions correctly? (b) What is the probability that the student answers fewer than 5 questions correctly?

a) 8*88*10⁻⁶ (0.00088 %)

b) 0.2137 (21.37%)

Step-by-step explanation:

if the test contains 25 questions and each questions is independent of the others, then the random variable X = answer "x" questions correctly, has a binomial probability distribution. Then

P (X=x) = n! / ((n-x) !*x!) * p^x * (1-p) ^ (n-x)

where

n = total number of questions = 25

p = probability of getting a question right = 1/4

then

a) P (x=n) = p^n = (1/4) ²⁵ = 8*88*10⁻⁶ (0.00088 %)

b) P (x<5) = F (5)

where F (x) is the cumulative binomial probability distribution - Then from tables

P (x<5) = F (5) = 0.2137 (21.37%)