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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Binomial Probability Distribution", "Cumulative Binomial Probability" ], "new_concepts": [], "current_concepts": [ "Binomial Probability Distribution", "Cumulative Binomial Probability" ] } ## Identify the distribution parameters Using the *Binomial Probability Distribution* knowledge point $$ \begin{aligned} &n = 7\ &p = 0.3\ &q = 1 - p = 0.7 \end{aligned} $$ ## Express the target probability Using the *Cumulative Binomial Probability* knowledge point $$ \begin{aligned} P(x > 2) &= 1 - P(x \le 2)\ &= 1 - [P(x = 0) + P(x = 1) + P(x = 2)] \end{aligned} $$ ## Calculate individual probabilities Using the *Binomial Probability Distribution* knowledge point $$ \begin{aligned} P(x = 0) &= \binom{7}{0} (0.3)^0 (0.7)^7 = 1 \times 1 \times 0.0823543 = 0.0823543\ P(x = 1) &= \binom{7}{1} (0.3)^1 (0.7)^6 = 7 \times 0.3 \times 0.117649 = 0.2470629\ P(x = 2) &= \binom{7}{2} (0.3)^2 (0.7)^5 = 21 \times 0.09 \times 0.16807 = 0.3176523 \end{aligned} $$ ## Sum the complementary probabilities Using the *Cumulative Binomial Probability* knowledge point $$ \begin{aligned} P(x \le 2) &= 0.0823543 + 0.2470629 + 0.3176523\ &= 0.6470695 \end{aligned} $$ ## Compute the final probability Using the *Cumulative Binomial Probability* knowledge point $$ \begin{aligned} P(x > 2) &= 1 - 0.6470695\ &= 0.3529305\ &\approx 0.353 \end{aligned} $$ $P(x > 2) \approx 0.353$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Statistics", "Cumulative Binomial Probability" ] }

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Explanation:

Identify the distribution parameters

Express the target probability

Calculate individual probabilities

Sum the complementary probabilities

Compute the final probability

Answer: