Can you pass a test by guessing? Solved step by step

Problem. A test has 20 multiple-choice questions with 4 options each, and you pass with at least 10 right. If you guess every answer at random, with exactly one correct option per question and each guess independent, what is the chance of passing?

Step by step

1. Each guess is right with probability p = 1/4 = 0.25 and wrong with probability 0.75, independently of the other questions.

2. The number of right answers X follows a binomial distribution with n = 20 and p = 0.25.

3. The average score from guessing is 20 × 0.25 = 5 right, which is only half of the 10 needed to pass.

4. The probability of exactly k right is C(20,k) × (0.25)^k × (0.75)^(20−k), where C(20,k) counts the ways to choose which k questions are right.

5. Passing means 10 or more right, so we add these terms for k = 10, 11, ..., 20: P(X ≥ 10) = sum of C(20,k) × (0.25)^k × (0.75)^(20−k).

6. Adding those eleven terms gives P(X ≥ 10) ≈ 0.0139, which is about 1.4%, or roughly 1 chance in 72.

7. Getting all 20 right needs every guess to be correct: 0.25^20 ≈ 9.1 × 10^-13, which is about 1 in 1.1 trillion.

Answer: Guessing gives about a 1.4% chance of passing (10 or more right out of 20), and about a 1 in 1.1 trillion chance of getting all 20 right.

This assumes pure random guessing on every question with exactly 4 equally likely options; real students can often rule out wrong options, which raises the odds.