Is the following a binomial scenario? Tommy has just completed a 20 question multiple choice quiz in his history class. Each question has four answers of which only one is correct. Since Tommy has not attended class this semester and has not done any of the reading, he has to guess on every question. Assuming he guesses on all 20 questions, what is the probability that he will pass the quiz (earn a grade of 60% or better)?单项选择题
A
Not binomial
B
Binomial

Question

Is the following a binomial scenario? Tommy has just completed a 20 question multiple choice quiz in his history class. Each question has four answers of which only one is correct. Since Tommy has not attended class this semester and has not done any of the reading, he has to guess on every question. Assuming he guesses on all 20 questions, what is the probability that he will pass the quiz (earn a grade of 60% or better)?单项选择题

A

Not binomial

B

Binomial

BlackTom AI · Accepted Answer

1. Relevant knowledge points and concepts
A binomial experiment meets four conditions: (i) a fixed number n of trials, (ii) each trial has exactly two possible outcomes (success/failure), (iii) trials are independent, and (iv) the probability of success p is constant across trials. The binomial probability for exactly k successes is P(X = k) = C(n,k) p^k (1−p)^(n−k).

2. Complete reasoning process step by step
- Identify trials: each of the 20 multiple-choice questions is a trial, so n = 20 (fixed).
- Define success: "correct answer" is success; failure is "incorrect answer" — exactly two outcomes per trial.
- Probability of success: guessing randomly among four choices gives p = 1/4 = 0.25, constant for every question.
- Independence: Tommy’s guess on one question does not affect his guess on another, so trials can be treated as independent.
All four binomial conditions are satisfied.

3. Calculations and intermediate steps (setup)
- Passing threshold: 60% of 20 = 0.60 × 20 = 12, so passing requires at least 12 correct answers.
- Random variable X ~ Binomial(n = 20, p = 0.25).
- Probability of passing: P(X ≥ 12) = sum_{k=12}^{20} C(20,k) (0.25)^k (0.75)^(20−k).
Example term for k = 12: C(20,12) = C(20,8) = 125,970, so the k=12 term is 125,970 × (0.25)^12 × (0.75)^8. (Full numeric summation is straightforward but not required to classify the scenario.)

4. Why the correct answer is correct
Each of the binomial criteria holds: fixed n, two outcomes per trial, independent trials, and constant success probability p = 0.25. Therefore the situation is properly modeled by a binomial distribution.

5. Why the other option is incorrect
"Not binomial" would apply only if one of the four criteria failed (e.g., variable p, dependent trials, non-binary outcomes, or non-fixed n). None of these issues occur here, so "Not binomial" is incorrect.

6. Conclusion confirming the final answer
All reasoning confirms that the scenario is Binomial. This matches the provided correct answer: Binomial.

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