Multiple choice test has 10 questions each with 5 choices 15

Multiple choice test has 10 questions, each with 5 choices. 1/5 choice is correct. An individual randomly guesses answer to every question. (Independent)

What is probability that individual:

Has two correct guesses at most?

Has 5 correct guesses?

Include all steps and formulas used to solve.

Solution

If there are n questions on the test, then the probability of answering exactly k correctly, if answers are chosen at random, is

(nck) * (1/5)^k * (4/5)^(n-k)

a. Case 1, k = 0 , Case 2, K=1 , Case 3, K=2

Hence, (10c0) * (4/5)^10 + (10c1)*(1/5)*(4/5)^9 + (10c2) * (1/5)^2*(4/5)^8

0.107 + 0.268 + 0.301 = 0.676

b. here, n= 10, k=5

10c5 * (1/5)^5 * (4/5)^5 = 0.026