a multiple-choice test consists of 7 questions. each question has answer choices of a, b, c, d, and e, and only one of the choices is correct. if a student randomly guesses on each question, what is the probability that she gets at most 1 of them correct?

carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

(if necessary, consult a list of formulas.)

Question

a multiple-choice test consists of 7 questions. each question has answer choices of a, b, c, d, and e, and only one of the choices is correct. if a student randomly guesses on each question, what is the probability that she gets at most 1 of them correct?

carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

(if necessary, consult a list of formulas.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Binomial Probability Formula", "Binomial Distribution" ], "new_concepts": [], "current_concepts": [ "Binomial Probability Formula", "Binomial Distribution" ] } ## Define the binomial parameters Using the *Binomial Distribution* knowledge point $$ \begin{aligned} &n = 7\ &p = \frac{1}{5} = 0.2\ &q = 1 - p = 0.8\ &X \sim \text{Binomial}(n = 7, p = 0.2) \end{aligned} $$ ## Calculate the probability of getting exactly 0 correct Using the *Binomial Probability Formula* knowledge point $$ \begin{aligned} P(X = 0) &= \binom{7}{0} (0.2)^0 (0.8)^7\ &= 1 \times 1 \times 0.2097152\ &= 0.2097152 \end{aligned} $$ ## Calculate the probability of getting exactly 1 correct Using the *Binomial Probability Formula* knowledge point $$ \begin{aligned} P(X = 1) &= \binom{7}{1} (0.2)^1 (0.8)^6\ &= 7 \times 0.2 \times 0.262144\ &= 0.3670016 \end{aligned} $$ ## Sum the probabilities for at most 1 correct Using the *Binomial Distribution* knowledge point $$ \begin{aligned} P(X \le 1) &= P(X = 0) + P(X = 1)\ &= 0.2097152 + 0.3670016\ &= 0.5767168 \end{aligned} $$ ## Round the final probability to two decimal places Using the *Binomial Distribution* knowledge point $$ \begin{aligned} P(X \le 1) \approx 0.58 \end{aligned} $$ A multiple-choice test consists of 7 questions. Each question has answer choices of a, b, c, d, and e, and only one of the choices is correct. If a student randomly guesses on each question, what is the probability that she gets at most 1 of them correct? Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places. 0.58 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Binomial Distribution" ] }

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QUESTION IMAGE

Question

Explanation:

Define the binomial parameters

Calculate the probability of getting exactly 0 correct

Calculate the probability of getting exactly 1 correct

Sum the probabilities for at most 1 correct

Round the final probability to two decimal places

Answer: