assume the random variable $x$ has a binomial distribution with the given probability of obtaining a success. find the following probability, given the number of trials and the probability of obtaining a success. round your answer to four decimal places.

\p(x \le 3), n = 7, p = 0.2\

Question

assume the random variable $x$ has a binomial distribution with the given probability of obtaining a success. find the following probability, given the number of trials and the probability of obtaining a success. round your answer to four decimal places.

\p(x \le 3), n = 7, p = 0.2\

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 Distribution", "Cumulative Binomial Probability" ], "new_concepts": [], "current_concepts": [ "Binomial Distribution", "Cumulative Binomial Probability" ] } ## Identify parameters Using the *Binomial Distribution* knowledge point $$ \begin{aligned} &n = 7\ &p = 0.2\ &q = 1 - p = 0.8\ &P(X = k) = \binom{7}{k} (0.2)^k (0.8)^{7-k} \end{aligned} $$ ## Calculate individual probabilities Using the *Binomial Distribution* knowledge point $$ \begin{aligned} P(X = 0) &= \binom{7}{0} (0.2)^0 (0.8)^7 = 1 \cdot 1 \cdot 0.2097152 = 0.2097152\ P(X = 1) &= \binom{7}{1} (0.2)^1 (0.8)^6 = 7 \cdot 0.2 \cdot 0.262144 = 0.3670016\ P(X = 2) &= \binom{7}{2} (0.2)^2 (0.8)^5 = 21 \cdot 0.04 \cdot 0.32768 = 0.2752512\ P(X = 3) &= \binom{7}{3} (0.2)^3 (0.8)^4 = 35 \cdot 0.008 \cdot 0.4096 = 0.114688 \end{aligned} $$ ## Sum the probabilities Using the *Cumulative Binomial Probability* knowledge point $$ \begin{aligned} P(X \le 3) &= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)\ P(X \le 3) &= 0.2097152 + 0.3670016 + 0.2752512 + 0.114688\ P(X \le 3) &= 0.966656 \approx 0.9667 \end{aligned} $$ Assume the random variable $X$ has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places. $P(X \le 3), n = 7, p = 0.2$ Answer: 0.9667 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Cumulative Binomial Probability" ] }

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QUESTION IMAGE

Question

Explanation:

Identify parameters

Calculate individual probabilities

Sum the probabilities

Answer: