Statistics Probability function

Question

A test consists of 5 questions. there are 4 answers for each question of which only one is correct. Assume that a student randomly guesses each question.

1. determine the probability function for X, that is p(x)
2. what is the probability that only three questions are answered correctly
3. what is the probability of at most 3 questions are answered correctly

Answer 1

probability is 0.25 that the student gets each answer correct and 0.75 that they get it wrong. this is a binomial distribution.

x P(x)

0 (0.75)^5= 0.237

1 5 * (0.25) (0.75)^4= 0.396

2 10 * (0.25)^2 (0.75)^3 = 0.264

3 10 * (0.25)^3 * (0.75)^2 = 0.088

4 5 * (0.25)^4 * 0.75 = 0.015

5 (0.25)^5 = 0.001

probability only answer 3 is 0.088

probability at most 3 = 0.237 + 0.396 + 0.264 + 0.088 = 0.985