Question

Suppose, you are attending an exam where there are 9 multiple choice questions with 3 options in each. A question can have multiple correct answers, and in that case, you have to check all of the correct answers to get the question right. a. What is the probability that you'll get exactly 6 of the questions right? b. What is the probability that you'll get at most 6 of the questions right?

Answer 1

a. The probability of getting exactly 6 of the question right is approximately 0.164

b. The probability of getting at most 6 of the questions right is approximately 0.93

Explanation:

The given parameters are;

The number of multiple choice questions = 9 multiple choice questions

The number of options in each question = 3 options

The number of outcomes to the answers of the question = 2, Right or Wrong

Therefore, the probability of having a question right, p = 1/2

The probability of having a question wrong, q = 1 - p = 1 - 1/2 = 1/2

By binomial probability theorem, we have;

The probability of getting exactly 6 of the question right, P(6), is given as follows;

P(6) = ₉C₆·p⁶·(q)⁹⁻⁶ = 84 × (1/2)⁶ × (1/2)³ = 0.1640625

The probability of getting exactly 6 of the question right, P(6) = 0.1640625 ≈ 0.164

b. The probability of P(at most 6 of the questions right) = P(X ≤ 6)

P(at most 6 of the questions right) = P(X ≤ 6) = 1 - P(X ≥ 7)

∴ P(X ≤ 6) = 1 - (₉C₇·p⁷·q⁹⁻⁷) = 1 -(36·(1/2)⁷·(1/2)² = 0.9299875

The probability of P(at most 6 of the questions right) = P(X ≤ 6) = 0.9299875 ≈ 0.93.