an automobile manufacturer finds that 1 in every 2000 automobiles produced has a particular manufacturing defect.
(a) use a binomial distribution to find the probability of finding 4 cars with the defect in a random sample of 7000 cars.
(b) the poisson distribution can be used to approximate the binomial distribution for large values of n and small values of p. repeat (a) using a poisson distribution and compare the results.

(a) the probability using the binomial distribution is .
(round to five decimal places as needed.)

Question

an automobile manufacturer finds that 1 in every 2000 automobiles produced has a particular manufacturing defect.
(a) use a binomial distribution to find the probability of finding 4 cars with the defect in a random sample of 7000 cars.
(b) the poisson distribution can be used to approximate the binomial distribution for large values of n and small values of p. repeat (a) using a poisson distribution and compare the results.

(a) the probability using the binomial distribution is  .
(round to five decimal places as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Binomial Probability Formula" ], "new_concepts": [], "current_concepts": [ "Binomial Probability Formula", "Poisson Distribution", "Binomial vs Geometric" ] } ## Identify parameters for the binomial distribution $$ \begin{aligned} p &= \frac{1}{2000} = 0.0005 \ q &= 1 - p = 0.9995 \ n &= 7000 \ k &= 4 \end{aligned} $$ ## Apply the binomial probability formula $$ \begin{aligned} P(X = 4) &= \binom{7000}{4} (0.0005)^4 (0.9995)^{6996} \ \binom{7000}{4} &= \frac{7000 \times 6999 \times 6998 \times 6997}{24} \approx 1.00196 \times 10^{14} \ (0.0005)^4 &= 6.25 \times 10^{-14} \ (0.9995)^{6996} &\approx e^{-7000 \times 0.0005} = e^{-3.5} \approx 0.03019738 \ P(X = 4) &\approx 1.00196 \times 10^{14} \times 6.25 \times 10^{-14} \times 0.03011204 \approx 0.18857 \end{aligned} $$ ## Round to five decimal places $$ P(X = 4) \approx 0.18857 $$ (a) The probability using the binomial distribution is 0.18857 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Binomial Probability Formula" ] }

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Question

Explanation:

Identify parameters for the binomial distribution

Apply the binomial probability formula

Round to five decimal places

Answer: