NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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A school in Brisbane is ordering new art supplies. The order includes 60 sketchbooks and 31 coloured pencils.

Two items are selected at random without replacement to check quality. List all possible outcomes for the item types using a table.

What is the exact probability that the first item selected is a sketchbook and the second is a coloured pencil? Express your answer as a simplified fraction.

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Two independent fair coin tosses. HH, HT, TH and TT are equally likely. Two outcomes have exactly one head: probability = 2/4 = 1/2.HHHT✓TH✓TTExactly one head: HT or TH
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Worked solution

  1. The school has 60 sketchbooks and 31 coloured pencils, so the total number of items is 60 + 31 = 91.
  2. Two items are selected at random without replacement, so the selection is dependent. The first item affects the probability of the second.
  3. We are to list all possible outcomes for the item types using a table. The possible outcomes are: (Sketchbook, Sketchbook), (Sketchbook, Coloured Pencil), (Coloured Pencil, Sketchbook), (Coloured Pencil, Coloured Pencil).
  4. We are to find the probability that the first item is a sketchbook and the second is a coloured pencil. This is a compound event with dependence.
  5. The probability that the first item is a sketchbook is 60/91, since there are 60 sketchbooks out of 91 total items.
  6. After selecting one sketchbook, there are now 59 sketchbooks and 31 coloured pencils left, so 90 items remain.
  7. The probability that the second item is a coloured pencil, given the first was a sketchbook, is 31/90.
  8. The combined probability is found by multiplying the two probabilities: (60/91) × (31/90).
  9. Calculate the product: (60 × 31) / (91 × 90) = 1860 / 8190.
  10. Simplify the fraction 1860/8190. First, divide numerator and denominator by 30: 1860 ÷ 30 = 62, 8190 ÷ 30 = 273. So 62/273.
  11. Check if 62 and 273 have any common factors. 62 = 2 × 31, 273 = 3 × 7 × 13. No common factors, so 62/273 is in simplest form.
  12. Therefore, the exact probability that the first item is a sketchbook and the second is a coloured pencil is 62/273.

Reference answer: 62/273

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