a warehouse employs 22 workers on first shift, 17 workers on second shift, and 13 workers on third shift. eight workers are chosen at random to be interviewed about the work environment. find the probability of choosing exactly two second shift workers and two third shift workers.

the probability of choosing two second shift workers and two third shift workers is .
(round to three decimal places as needed.)

Question

a warehouse employs 22 workers on first shift, 17 workers on second shift, and 13 workers on third shift. eight workers are chosen at random to be interviewed about the work environment. find the probability of choosing exactly two second shift workers and two third shift workers.

the probability of choosing two second shift workers and two third shift workers is .
(round to three decimal places as needed.)

the probability of choosing two second shift workers and two third shift workers is  .
(round to three decimal places as needed.)

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": [ "Combinatorics", "Sampling Without Replacement", "Hypergeometric Distribution" ], "new_concepts": [], "current_concepts": [ "Combinatorics", "Sampling Without Replacement", "Hypergeometric Distribution", "Classical Probability" ] } ## Define the population and sample sizes Using the *Combinatorics* knowledge point $$ \begin{aligned} &\text{First shift workers } (N_1) = 22\ &\text{Second shift workers } (N_2) = 17\ &\text{Third shift workers } (N_3) = 13\ &\text{Total workers } (N) = 22 + 17 + 13 = 52\ &\text{Sample size } (n) = 8 \end{aligned} $$ ## Determine the required selection counts Using the *Sampling Without Replacement* knowledge point $$ \begin{aligned} &\text{Second shift workers to choose } (k_2) = 2\ &\text{Third shift workers to choose } (k_3) = 2\ &\text{First shift workers to choose } (k_1) = n - k_2 - k_3 = 8 - 2 - 2 = 4 \end{aligned} $$ ## Calculate the total number of possible samples Using the *Combinatorics* knowledge point $$ \begin{aligned} \text{Total outcomes } = \binom{52}{8} = \frac{52!}{8!(52-8)!} = 752,538,150 \end{aligned} $$ ## Calculate the number of favorable outcomes Using the *Hypergeometric Distribution* knowledge point $$ \begin{aligned} \text{Favorable outcomes } &= \binom{22}{4} \times \binom{17}{2} \times \binom{13}{2}\ \binom{22}{4} &= \frac{22 \times 21 \times 20 \times 19}{4 \times 3 \times 2 \times 1} = 7,315\ \binom{17}{2} &= \frac{17 \times 16}{2 \times 1} = 136\ \binom{13}{2} &= \frac{13 \times 12}{2 \times 1} = 78\ \text{Favorable outcomes } &= 7,315 \times 136 \times 78 = 77,601,480 \end{aligned} $$ ## Calculate the probability Using the *Hypergeometric Distribution* knowledge point $$ \begin{aligned} P &= \frac{77,601,480}{752,538,150} \approx 0.10311966\ P &\approx 0.103 \quad (\text{rounded to three decimal places}) \end{aligned} $$ The probability of choosing two second shift workers and two third shift workers is 0.103 (Round to three decimal places as needed.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Hypergeometric Distribution" ] }

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QUESTION IMAGE

Question

Explanation:

Define the population and sample sizes

Determine the required selection counts

Calculate the total number of possible samples

Calculate the number of favorable outcomes

Calculate the probability

Answer: