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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Binomial Probability Distribution" ], "new_concepts": [], "current_concepts": [ "Binomial Probability Distribution" ] } ## Identify the binomial parameters Using the *Binomial Probability Distribution* knowledge point $$ \begin{aligned} &n = 7\ &p = 0.8\ &q = 1 - p = 0.2\ &k = 3 \end{aligned} $$ ## Apply the binomial formula Using the *Binomial Probability Distribution* knowledge point $$ \begin{aligned} P(x = 3) &= \binom{7}{3} \cdot (0.8)^3 \cdot (0.2)^{7-3}\ &= \frac{7!}{3!4!} \cdot (0.8)^3 \cdot (0.2)^4 \end{aligned} $$ ## Calculate the combinations and powers Using the *Binomial Probability Distribution* knowledge point $$ \begin{aligned} \binom{7}{3} &= \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35\ (0.8)^3 &= 0.512\ (0.2)^4 &= 0.0016 \end{aligned} $$ ## Compute the final probability Using the *Binomial Probability Distribution* knowledge point $$ \begin{aligned} P(x = 3) &= 35 \cdot 0.512 \cdot 0.0016\ &= 17.92 \cdot 0.0016\ &= 0.028672\ &\approx 0.029 \end{aligned} $$ Consider a binomial random variable with $n = 7$ and $p = 0.8$. Let $x$ be the number of successes in the sample. Evaluate the probability. (Round your answer to three decimal places.) $P(x = 3) =$ $0.029$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Binomial Probability Distribution" ] }

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Question

Explanation:

Identify the binomial parameters

Apply the binomial formula

Calculate the combinations and powers

Compute the final probability

Answer: