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

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 < 5), n = 9, p = 0.6\

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 the distribution parameters Using the *Binomial Distribution* knowledge point $$ X \sim \text{Binomial}(n = 9, p = 0.6) $$ ## Formulate the cumulative probability Using the *Cumulative Binomial Probability* knowledge point $$ P(X < 5) = P(X \le 4) = \sum_{k=0}^{4} \binom{9}{k} (0.6)^k (0.4)^{9-k} $$ ## Calculate individual term probabilities Using the *Binomial Distribution* knowledge point $$ \begin{aligned} P(X = 0) &= \binom{9}{0} (0.6)^0 (0.4)^9 = 1 \cdot 1 \cdot 0.000262144 = 0.000262144 \ P(X = 1) &= \binom{9}{1} (0.6)^1 (0.4)^8 = 9 \cdot 0.6 \cdot 0.00065536 = 0.003538944 \ P(X = 2) &= \binom{9}{2} (0.6)^2 (0.4)^7 = 36 \cdot 0.36 \cdot 0.0016384 = 0.021233664 \ P(X = 3) &= \binom{9}{3} (0.6)^3 (0.4)^6 = 84 \cdot 0.216 \cdot 0.004096 = 0.074317824 \ P(X = 4) &= \binom{9}{4} (0.6)^4 (0.4)^5 = 126 \cdot 0.1296 \cdot 0.01024 = 0.167215104 \end{aligned} $$ ## Sum the probabilities and round Using the *Cumulative Binomial Probability* knowledge point $$ \begin{aligned} P(X \le 4) &= 0.000262144 + 0.003538944 + 0.021233664 + 0.074317824 + 0.167215104 \ &= 0.26656768 \ &\approx 0.2666 \end{aligned} $$ 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. $P(X < 5), n = 9, p = 0.6$ Answer: 0.2666 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Cumulative Binomial Probability" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the distribution parameters

Formulate the cumulative probability

Calculate individual term probabilities

Sum the probabilities and round

Answer: