a quality control inspector has drawn a sample of 12 light bulbs from a recent production lot. suppose 30% of the bulbs in the lot are defective. what is the probability that between 3 and 5 (both inclusive) bulbs from the sample are defective? round your answer to four decimal places.

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{ "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 Distribution" ], "new_concepts": [], "current_concepts": [ "Binomial Distribution", "Cumulative Binomial Probability" ] } ## Identify the distribution parameters Using the *Binomial Distribution* knowledge point $$ \begin{aligned} &n = 12\ &p = 0.30\ &q = 1 - p = 0.70 \end{aligned} $$ ## Define the probability of interest Using the *Binomial Distribution* knowledge point $$ \begin{aligned} &P(3 \le X \le 5) = P(X=3) + P(X=4) + P(X=5)\ &P(X=k) = \binom{n}{k} p^k q^{n-k} \end{aligned} $$ ## Calculate individual probabilities Using the *Binomial Distribution* knowledge point $$ \begin{aligned} P(X=3) &= \binom{12}{3} (0.3)^3 (0.7)^9 = 220 \times 0.027 \times 0.040353607 \approx 0.239700\ P(X=4) &= \binom{12}{4} (0.3)^4 (0.7)^8 = 495 \times 0.0081 \times 0.05764801 \approx 0.231140\ P(X=5) &= \binom{12}{5} (0.3)^5 (0.7)^7 = 792 \times 0.00243 \times 0.0823543 \approx 0.158496 \end{aligned} $$ ## Sum the probabilities and round Using the *Binomial Distribution* knowledge point $$ \begin{aligned} P(3 \le X \le 5) &\approx 0.239700 + 0.231140 + 0.158496 = 0.629336\ P(3 \le X \le 5) &\approx 0.6293 \end{aligned} $$ A quality control inspector has drawn a sample of 12 light bulbs from a recent production lot. Suppose 30% of the bulbs in the lot are defective. What is the probability that between 3 and 5 (both inclusive) bulbs from the sample are defective? Round your answer to four decimal places. 0.6293 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Binomial Distribution" ] }

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Explanation:

Identify the distribution parameters

Define the probability of interest

Calculate individual probabilities

Sum the probabilities and round

Answer: