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Problem scenario:
- University information technology office received 20 service orders for printer problems:
- 8 laser printers
- 12 inkjet printers
- A sample of 5 service orders is to be selected for a customer satisfaction survey.
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Selection process:
- The 5 service orders are selected randomly.
- Any subset of size 5 has the same chance of being selected as any other subset.
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Objective:
- Determine the probability of exactly service orders for inkjet printers in the sample, where .
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Key points:
- Total service orders: 20
- Service orders for laser printers: 8
- Service orders for inkjet printers: 12
- Sample size: 5
- Calculate probability for various values of (number of inkjet printers selected in the sample).
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Population details:
- Total population size:
- Sample size:
- Number of inkjet printers (denoted ):
- Number of laser printers (denoted ):
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Case where :
- We are interested in the probability that exactly 2 inkjet printers are selected in the sample.
- Since all outcomes (each consisting of 5 specific orders) are equally likely, the probability can be calculated based on combinatorics.
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Probability setup:
- The total number of ways to choose 5 orders from 20:
- The number of ways to choose exactly 2 inkjet printers from 12:
- The number of ways to choose the remaining 3 printers from the 8 laser printers:
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Probability expression:
- The probability for exactly inkjet printers in the sample:
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Total number of possible outcomes:
- The number of ways to select 5 service orders from 20 without regard to order is:
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Number of outcomes where (exactly 2 inkjet printers are selected):
- The number of ways to select 2 inkjet orders from the 12 inkjet printers:
- For each of these selections, the number of ways to select the remaining 3 orders from the 8 laser printers:
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Applying the product rule:
- The total number of outcomes with (2 inkjet printers and 3 laser printers) is given by:
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Probability expression:
- The probability that exactly 2 inkjet printers are selected is: