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

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In a simulation of a walking route in Melbourne, two paths are available from the train station to the city library. Path A is 810 m long and has a 60% chance of being blocked by construction on any given day.

Path B is 500 m long and has a 40% chance of being blocked. If a walker chooses a path at random and finds it blocked, they immediately switch to the other path.

What is the exact probability that a randomly selected walker who starts on Path A and ends up using Path B actually completes their journey via Path B, given that Path A was blocked on the day of travel? Express your answer as a simplified fraction.

Concept illustrations

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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.
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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: Path A is 810 m with a 60% chance of being blocked. Path B is 500 m with a 40% chance of being blocked. A walker starts on Path A and switches to Path B if Path A is blocked. We are to find the probability that the walker completes the journey via Path B, given that Path A was blocked.
  2. The event 'Path A was blocked' is given. We are to compute the conditional probability that the walker uses Path B and completes the journey, given this blockage.
  3. Since the walker switches to Path B immediately upon finding Path A blocked, and Path B has a 40% chance of being blocked, the journey via Path B only completes if Path B is NOT blocked.
  4. The probability that Path B is not blocked is 1 - 0.4 = 0.6.
  5. The event 'Path A blocked' is independent of 'Path B blocked'. Therefore, the conditional probability that Path B is not blocked given Path A is blocked is simply the probability that Path B is not blocked.
  6. Thus, the probability that the walker completes the journey via Path B, given Path A was blocked, is equal to the probability that Path B is not blocked, which is 0.6.
  7. Convert 0.6 to a simplified fraction: 0.6 = 6/10 = 3/5.
  8. The fraction 3/5 is already in simplest form, as 3 and 5 are coprime.
  9. Therefore, the exact probability is 3/5.
  10. Check: 3/5 = 0.6, which matches the probability that Path B is not blocked. The logic holds: if Path A is blocked, the walker uses Path B; they complete the journey only if Path B is not blocked.
  11. Final answer: the probability that the walker completes the journey via Path B, given Path A was blocked, is 3/5.
  12. The result is consistent with conditional probability: P(Path B not blocked | Path A blocked) = P(Path B not blocked) = 0.6 = 3/5.

Reference answer: 3/5

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