NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Conditional probability

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A school in Brisbane orders 82 science kits and 44 art kits for the year. Of the 82 science kits, 36 are for practical experiments and 46 are for theory.

Among the 44 art kits, 22 contain drawing materials and 22 contain painting materials. The school knows that 18 of the science kits contain both practical experiment components and drawing materials.

If a kit is selected at random and is known to contain drawing materials, what is the probability that it is a science kit? Express your answer as an exact fraction.

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Restrict the sample space before counting. All 12 dots are cyclists. Three filled dots also represent red clothing. Within the cyclist group, P(red given cycles) = 3/12 = 1/4.12 cyclists · 3 wearing red
All 12 dots are cyclists. Three filled dots also represent red clothing. Within the cyclist group, P(red given cycles) = 3/12 = 1/4.
A fair-coin simulation model. Choose 0 or 1 with equal probability: 0 means tails and 1 means heads. Repeat independent trials and count outcomes. No experiment results are asserted here.Choose 0 or 1Equal chance→0=T 1=HMap outcome→RepeatCount heads
Choose 0 or 1 with equal probability: 0 means tails and 1 means heads. Repeat independent trials and count outcomes. No experiment results are asserted here.

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Worked solution

  1. The school has 82 science kits and 44 art kits, totaling 126 kits. Among these, 18 science kits contain both practical experiment components and drawing materials.
  2. We are told that 22 art kits contain drawing materials. Since only science kits can have both practical experiment components and drawing materials, the 18 kits with both are part of the 22 drawing materials kits in art.
  3. The total number of kits that contain drawing materials is the sum of art kits with drawing materials (22) and science kits with drawing materials. However, 18 science kits are already counted in the drawing materials group, so we must determine how many science kits contain drawing materials.
  4. We are not told directly how many science kits contain drawing materials. But we know that 18 science kits contain both practical experiment components and drawing materials. Since drawing materials are only in art kits and some science kits, and 18 science kits have drawing materials, we assume that the 18 kits are part of the science kits that contain drawing materials.
  5. Let D be the set of kits containing drawing materials. |D| = 22 (art kits with drawing materials) + x (science kits with drawing materials). We are told that 18 science kits contain both practical components and drawing materials, so x ≥ 18.
  6. Since drawing materials are only in art kits (22) and some science kits, and 18 science kits contain drawing materials, the total number of kits with drawing materials is 22 (art) + 18 (science) = 40.
  7. We are to find the probability that a randomly selected kit containing drawing materials is a science kit. This is a conditional probability: P(science | drawing materials).
  8. By definition of conditional probability: P(A|B) = P(A and B) / P(B). Here, A = science kit, B = drawing materials. So P(science | drawing materials) = (number of science kits with drawing materials) / (total number of kits with drawing materials).
  9. Number of science kits with drawing materials = 18 (given). Total kits with drawing materials = 22 (art) + 18 (science) = 40.
  10. So P(science | drawing materials) = 18 / 40. Simplify the fraction: divide numerator and denominator by 2 → 9 / 20.
  11. The probability is 9/20. This is an exact fraction, as required.
  12. Final answer: 9/20. This means that given a kit contains drawing materials, there is a 9 in 20 chance it is a science kit.

Reference answer: 9/20

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