NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Data collection and sampling

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Required skills (click to reveal)

A craft designer in Melbourne is creating a large wall mural using 11 identical wooden panels and 34 identical fabric panels. To test the durability of the materials, the designer plans to randomly select 5 panels from the total of 45 panels (11 + 34) for a stress test.

What is the exact probability that exactly 2 of the selected panels are wooden and 3 are fabric? Express your answer as a simplified fraction.

Concept illustrations

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A sample is part of a population. The large outline represents a population of 24 members. Six filled members illustrate a sample. This picture does not establish that a selection process was random.6 selected from 24
The large outline represents a population of 24 members. Six filled members illustrate a sample. This picture does not establish that a selection process was random.
Median, mode and range in sorted data. The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.23389↑ medianmode 3 range 7
The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The total number of panels is 11 wooden + 34 fabric = 45 panels.
  2. The designer selects 5 panels at random from 45, so the total number of possible combinations is C(45, 5).
  3. We want exactly 2 wooden panels from 11, so the number of ways to choose 2 wooden panels is C(11, 2).
  4. We want exactly 3 fabric panels from 34, so the number of ways to choose 3 fabric panels is C(34, 3).
  5. The number of favourable outcomes is C(11, 2) × C(34, 3).
  6. Calculate C(11, 2) = 11! / (2! × (11-2)!) = (11 × 10) / (2 × 1) = 55.
  7. Calculate C(34, 3) = 34! / (3! × (34-3)!) = (34 × 33 × 32) / (3 × 2 × 1) = 5984.
  8. Multiply favourable outcomes: 55 × 5984 = 329,120.
  9. Calculate total possible outcomes: C(45, 5) = 45! / (5! × 40!) = (45 × 44 × 43 × 42 × 41) / (5 × 4 × 3 × 2 × 1) = 1,221,759.
  10. The probability is 329,120 / 1,221,759.
  11. Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD). GCD(329120, 1221759) = 1, so the fraction is already in simplest form.
  12. The exact probability that exactly 2 wooden and 3 fabric panels are selected is 329120/1221759.

Reference answer: 329120/1221759

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