a new surgery is successful 80% of the time. if the results of 10 such surgeries are randomly sampled, what is the probability that at least 9 of them are successful?

carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

(if necessary, consult a list of formulas.)

Question

a new surgery is successful 80% of the time. if the results of 10 such surgeries are randomly sampled, what is the probability that at least 9 of them are successful?

carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

(if necessary, consult a list of formulas.)

Accepted Answer

{ "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 Probability Formula", "Binomial Distribution" ], "new_concepts": [], "current_concepts": [ "Binomial Distribution", "Binomial Probability Formula" ] } ## Define the random variable and parameters Using the *Binomial Distribution* knowledge point $$ \begin{aligned} &X \sim \text{Binomial}(n = 10, p = 0.8) \end{aligned} $$ ## State the probability formula Using the *Binomial Probability Formula* knowledge point $$ \begin{aligned} &P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \end{aligned} $$ ## Calculate the individual probabilities Using the *Binomial Probability Formula* knowledge point $$ \begin{aligned} &P(X = 9) = \binom{10}{9} (0.8)^9 (0.2)^1 = 10 \times 0.134217728 \times 0.2 = 0.268435456 \approx 0.2684\ &P(X = 10) = \binom{10}{10} (0.8)^{10} (0.2)^0 = 1 \times 0.1073741824 \times 1 = 0.1073741824 \approx 0.1074 \end{aligned} $$ ## Sum the probabilities for the final result Using the *Binomial Distribution* knowledge point $$ \begin{aligned} &P(X \ge 9) = P(X = 9) + P(X = 10)\ &P(X \ge 9) = 0.268435456 + 0.1073741824 = 0.3758096384 \approx 0.38 \end{aligned} $$ A new surgery is successful 80% of the time. If the results of 10 such surgeries are randomly sampled, what is the probability that at least 9 of them are successful? Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places. 0.38 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Binomial Distribution" ] }

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QUESTION IMAGE

Question

Explanation:

Define the random variable and parameters

State the probability formula

Calculate the individual probabilities

Sum the probabilities for the final result

Answer: