times student will get 3 out of 10 questions right randomly guessing 4 choice questions

Question

a student takes a multipl choice test that has 10 questions, each question has 4 choices, the student guesses randomly at each answer, what is the probability they will get 3 correct.

Answer 1

This is a binomial distribution, where p=1/4=0.25, the chance of getting the correct answer, and 1-p=0.75, the chance of getting the wrong answer. The distribution's last 4 terms are:

(1-p)^10, 10(1-p)^9p, 45(1-p)^8p^2, 120(1-p)^7p ^3.

These represent the chances of 0, 1, 2, 3 correct answers in 10. The coefficients are the number of ways of 0, 1, 2, 3 correct answers can be mixed with wrong answers. We need the last in this list which evaluates to 120*0.75^7*0.25^3=0.25 approx. and is the probability of getting exactly 3 right out of 10.