NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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A walking route in Melbourne offers two independent choices at a junction: a path to the lake (L) or a path to the forest (F), and a choice of direction at the next fork: left (X) or right (Y). The probability of choosing the lake path is 62/100, and the probability of choosing the left direction at the second fork is 58/100.

Using a two-way table to represent all possible combinations of the two events, determine the exact probability that a walker chooses the lake path AND the left direction. 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
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. The two independent events are: choosing the lake path (L) with probability 62/100, and choosing the left direction (X) at the second fork with probability 58/100.
  2. Since the two choices are independent, the probability of both occurring together is the product of their individual probabilities.
  3. Set up the multiplication: Pr(L and X) = Pr(L) × Pr(X) = (62/100) × (58/100).
  4. Multiply the numerators: 62 × 58 = 3596.
  5. Multiply the denominators: 100 × 100 = 10000.
  6. The combined probability is 3596/10000.
  7. Simplify the fraction by finding the greatest common divisor (GCD) of 3596 and 10000. First, factor both numbers.
  8. Factor 3596: divide by 2 → 1798, again → 899. 899 ÷ 29 = 31. So 3596 = 2² × 29 × 31.
  9. Factor 10000: 10000 = 10⁴ = (2×5)⁴ = 2⁴ × 5⁴.
  10. The common factors are 2² = 4. Divide numerator and denominator by 4: 3596 ÷ 4 = 899, 10000 ÷ 4 = 2500.
  11. The simplified fraction is 899/2500. Check that 899 and 2500 have no common factors: 899 = 29×31, 2500 = 2²×5⁴. No common prime factors.
  12. Therefore, the probability that a walker chooses the lake path AND the left direction is 899/2500.

Reference answer: 899/2500

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