If a person randomly guesses the answers to a 5-question true/false quiz, what is the probability of getting at least 4 questions correct?

Instruction: A five-question true-or-false test has one correct choice per question. A person independently guesses each answer with equal probability for true and false. What is the probability of getting at least four correct?

Context: This question assesses the candidate's understanding of binomial distributions and their application to a scenario with binary outcomes.

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Official answer

Let X be the number of correct answers. Under independent fair guessing, X has a binomial distribution with n = 5 and p = 1/2. “At least four” includes exactly four and exactly five:

P(X >= 4) = C(5,4)(1/2)^5 + C(5,5)(1/2)^5 = (5 + 1)/32 = 3/16 = 18.75%.

Equivalently, all 32 answer patterns are equally likely. Five patterns miss exactly one question, and one pattern gets everything right.

Assessment: Include the all-correct case. If guesses are not independent or are not equally likely to be correct, this binomial model needs to change.

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