NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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In a walking route competition in Sydney, participants must choose two different paths from a set of 78 walking trails, where 85 trails are marked as 'scenic' and 78 as 'moderate'. Two trails are selected at random without replacement.

Use a table or list to determine all possible outcomes for the trail types. What is the exact probability that the first trail selected is scenic and the second is moderate?

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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Worked solution

  1. Given: 85 scenic trails and 78 moderate trails, total trails = 85 + 78 = 163. Two trails are selected at random without replacement.
  2. We are to find the probability that the first trail is scenic and the second is moderate.
  3. The probability of selecting a scenic trail first is: P(first scenic) = 85 / 163.
  4. After selecting one scenic trail, 162 trails remain, with 78 still moderate. So P(second moderate | first scenic) = 78 / 162.
  5. The combined probability is the product: P = (85 / 163) * (78 / 162).
  6. Multiply numerators: 85 * 78 = 6630.
  7. Multiply denominators: 163 * 162 = 26406.
  8. So P = 6630 / 26406.
  9. Simplify the fraction by finding the greatest common divisor (GCD) of 6630 and 26406. First, factor both numbers.
  10. GCD(6630, 26406) = 6. Divide numerator and denominator by 6: 6630 ÷ 6 = 1105, 26406 ÷ 6 = 4401.
  11. The simplified fraction is 1105 / 4401. Check if further simplification is possible: 1105 and 4401 share no common factors.
  12. Therefore, the exact probability is 1105 / 4401. This matches the required form as a simplified fraction.

Reference answer: 1105/4401

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