find the indicated probabilities using the geometric distribution, the poisson distribution, or the binomial distribution. then determine if the events are unusual. if convenient, use the appropriate probability table or technology to find the probabilities.

the mean number of heart transplants performed per day in a country is about five. find the probability that the number of heart transplants performed on any given day is (a) exactly six, (b) at least eight, and (c) no more than four.

(a) p(6) = 0.146
(round to three decimal places as needed.)

(b) p(at least 8) =
(round to three decimal places as needed.)

Question

find the indicated probabilities using the geometric distribution, the poisson distribution, or the binomial distribution. then determine if the events are unusual. if convenient, use the appropriate probability table or technology to find the probabilities.

the mean number of heart transplants performed per day in a country is about five. find the probability that the number of heart transplants performed on any given day is (a) exactly six, (b) at least eight, and (c) no more than four.

(a) p(6) = 0.146
(round to three decimal places as needed.)

(b) p(at least 8) =
(round to three 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": [ "Poisson Distribution", "Poisson Probability Formula" ], "new_concepts": [], "current_concepts": [ "Poisson Distribution", "Poisson Probability Formula" ] } ## Identify the distribution and parameters $$ \begin{aligned} &\lambda = 5 \quad (\text{mean number of transplants per day})\ &X \sim \text{Poisson}(\lambda = 5)\ &P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} = \frac{e^{-5} 5^x}{x!} \end{aligned} $$ ## Calculate the probability of at least eight transplants $$ \begin{aligned} P(X \ge 8) &= 1 - P(X \le 7) = 1 - \sum_{x=0}^{7} \frac{e^{-5} 5^x}{x!}\ P(X = 0) &\approx 0.006738\ P(X = 1) &\approx 0.033690\ P(X = 2) &\approx 0.084224\ P(X = 3) &\approx 0.140374\ P(X = 4) &\approx 0.175467\ P(X = 5) &\approx 0.175467\ P(X = 6) &\approx 0.146223\ P(X = 7) &\approx 0.104445\ P(X \le 7) &\approx 0.006738 + 0.033690 + 0.084224 + 0.140374 + 0.175467 + 0.175467 + 0.146223 + 0.104445 = 0.866628\ P(X \ge 8) &\approx 1 - 0.866628 = 0.133372 \approx 0.133 \end{aligned} $$ (b) P(at least 8) = 0.133 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Poisson Probability Formula" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the distribution and parameters

Calculate the probability of at least eight transplants

Answer: