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{ "quality": "blurry", "question_count": 2, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Geometric Distribution", "Geometric Probability Formula" ], "new_concepts": [], "current_concepts": [ "Geometric Distribution", "Geometric Probability Formula" ] } ## Identify parameters of the geometric distribution $$ \begin{aligned} p &= \frac{1}{500} = 0.002 \ q &= 1 - p = 0.998 \end{aligned} $$ ## Calculate probability for part (b) $$ \begin{aligned} P(X \le 3) &= P(1) + P(2) + P(3) \ &= p + pq + pq^2 \ &= 0.002 + 0.002(0.998) + 0.002(0.998)^2 \ &= 0.002 + 0.001996 + 0.001992008 \ &= 0.005988008 \approx 0.006 \end{aligned} $$ ## Question 1 (a) P(the first warped glass item is the 10th item produced) = 0.002 ## Question 2 (b) P(the first warped item is the first, second, or third item produced) = 0.006 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Geometric Probability Formula" ] }

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Identify parameters of the geometric distribution

Calculate probability for part (b)

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Question 1

Question 2