a warehouse employs 23 workers on first shift, 15 workers on second shift, and 10 workers on third shift. eight workers are chosen at random to be interviewed about the work environment. find the probability of choosing exactly five first-shift workers.

the probability of choosing exactly five first-shift workers is .
(round to three decimal places as needed.)

Question

a warehouse employs 23 workers on first shift, 15 workers on second shift, and 10 workers on third shift. eight workers are chosen at random to be interviewed about the work environment. find the probability of choosing exactly five first-shift workers.

the probability of choosing exactly five first-shift workers is .
(round to three decimal places as needed.)

the probability of choosing exactly five first-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", "Hypergeometric Distribution" ], "new_concepts": [], "current_concepts": [ "Combinatorics", "Hypergeometric Distribution", "Classical Probability" ] } ## Identify the population parameters Using the **Hypergeometric Distribution** and **Combinatorics** knowledge points $$ \begin{aligned} &\text{First-shift workers } (S) = 23\ &\text{Other workers } (O) = 15 + 10 = 25\ &\text{Total workers } (N) = 23 + 25 = 48\ &\text{Sample size } (n) = 8\ &\text{Desired successes } (k) = 5 \end{aligned} $$ ## Calculate the number of successful outcomes Using the **Combinatorics** knowledge point $$ \begin{aligned} &\text{Ways to choose 5 first-shift workers: } \binom{23}{5} = \frac{23 \times 22 \times 21 \times 20 \times 19}{5 \times 4 \times 3 \times 2 \times 1} = 33,649\ &\text{Ways to choose 3 other workers: } \binom{25}{3} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = 2,300\ &\text{Total successful ways: } \binom{23}{5} \times \binom{25}{3} = 33,649 \times 2,300 = 77,392,700 \end{aligned} $$ ## Calculate the total number of possible outcomes Using the **Combinatorics** knowledge point $$ \begin{aligned} &\text{Total ways to choose 8 workers: } \binom{48}{8} = \frac{48 \times 47 \times 46 \times 45 \times 44 \times 43 \times 42 \times 41}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 377,348,994 \end{aligned} $$ ## Compute the probability Using the **Hypergeometric Distribution** knowledge point $$ \begin{aligned} &P(X = 5) = \frac{\binom{23}{5} \times \binom{25}{3}}{\binom{48}{8}} = \frac{77,392,700}{377,348,994} \approx 0.20510 \ &\text{Rounded to three decimal places: } 0.205 \end{aligned} $$ A warehouse employs 23 workers on first shift, 15 workers on second shift, and 10 workers on third shift. Eight workers are chosen at random to be interviewed about the work environment. Find the probability of choosing exactly five first-shift workers. The probability of choosing exactly five first-shift workers is 0.205. (Round to three decimal places as needed.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Hypergeometric Distribution" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify the population parameters

Calculate the number of successful outcomes

Calculate the total number of possible outcomes

Compute the probability

Answer: