NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Combined events and probability diagrams

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A craft designer in Melbourne is creating a pattern using two types of coloured tiles: 96 red tiles and 22 blue tiles. The designer randomly selects one tile from the combined set to place in a corner of the design.

What is the exact probability that the selected tile is red? Use a two-way table to determine the probability of this event, and express your answer as a simplified fraction.

Concept illustrations

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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
HH, HT, TH and TT are equally likely. Two outcomes have exactly one head: probability = 2/4 = 1/2.
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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Explanation

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Worked solution

  1. The total number of tiles is the sum of red and blue tiles: 96 red + 22 blue = 118 tiles.
  2. The probability of selecting a red tile is the number of red tiles divided by the total number of tiles: 96 / 118.
  3. To use a two-way table, define the two events: tile colour (red or blue) and selection outcome (selected or not selected). Since only one tile is selected, the table will show the counts for red and blue tiles in the selection and non-selection categories.
  4. Construct a two-way table with rows for 'Selected' and 'Not Selected', and columns for 'Red' and 'Blue'. The total for 'Selected' is 1 (since one tile is selected), and 'Not Selected' is 117.
  5. Fill in the table: under 'Selected', place 96 in the 'Red' column and 22 in the 'Blue' column. Under 'Not Selected', place 0 in 'Red' and 0 in 'Blue' because only one tile is selected and its colour is known.
  6. The two-way table now shows that the selected tile is red in 96 out of 118 possible outcomes.
  7. Simplify the fraction 96/118 by finding the greatest common divisor (GCD) of 96 and 118. The GCD is 2.
  8. Divide both numerator and denominator by 2: 96 ÷ 2 = 48, 118 ÷ 2 = 59. The simplified fraction is 48/59.
  9. Verify that 48 and 59 have no common factors: 59 is a prime number and does not divide 48, so the fraction is fully simplified.
  10. The probability that the selected tile is red is 48/59.
  11. Check: 48/59 ≈ 0.8136, and 96/118 ≈ 0.8136, so the simplification is correct.
  12. The exact probability that the selected tile is red is 48/59.

Reference answer: 48/59

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