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

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A craft designer in Brisbane creates a pattern using two overlapping shapes: a square and a circle. The square has side length 97 cm, and the circle has diameter 58 cm.

The circle is positioned so that its centre lies exactly on one corner of the square. What is the probability that a randomly selected point inside the square also lies inside the circle?

Express your answer as an exact fraction.

Circle measurementsdiameter = 58 cmd = 58 cm
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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 and answer (click to reveal)

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Worked solution

  1. The square has side length 97 cm, so its area is 97 cm × 97 cm.
  2. Calculate the area of the square: 97 × 97 = 9409 cm².
  3. The circle has diameter 58 cm, so its radius is 58 ÷ 2 = 29 cm.
  4. The area of a circle is π × r². Substitute r = 29 cm: π × 29².
  5. Calculate 29²: 29 × 29 = 841.
  6. So the area of the circle is π × 841 cm².
  7. The circle is positioned so its centre lies on one corner of the square. The square extends 97 cm in each direction from that corner.
  8. The circle has radius 29 cm, so it extends 29 cm from the corner in all directions. Since 29 < 97, the entire circle lies within the square along both axes from the corner.
  9. Therefore, the region of overlap between the square and the circle is exactly the area of the circle, because the circle lies entirely within the square from the corner.
  10. The probability that a randomly selected point inside the square also lies inside the circle is the ratio of the circle's area to the square's area: (π × 841) / 9409.
  11. Simplify the fraction: 841 / 9409. Note that 841 = 29² and 9409 = 97². So the fraction becomes (29²) / (97²) = (29/97)².
  12. The probability is (29/97)² = 841 / 9409. This is the exact fraction.

Reference answer: 841/9409

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