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 school's stationery order in Sydney, 55% of orders contain at least one pen, and 40% contain at least one pencil. The probability that an order contains both a pen and a pencil is 0.36.

Using a repeated chance simulation model, what is the exact conditional probability that an order contains a pencil, given that it contains a pen? Express your answer as a simplified fraction.

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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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. Given: Probability an order contains at least one pen, P(Pen) = 55% = 0.55.
  2. Given: Probability an order contains at least one pencil, P(Pencil) = 40% = 0.40.
  3. Given: Probability an order contains both a pen and a pencil, P(Pen ∩ Pencil) = 0.36.
  4. We are to find the conditional probability that an order contains a pencil given it contains a pen, which is P(Pencil | Pen).
  5. The formula for conditional probability is P(A|B) = P(A ∩ B) / P(B).
  6. Substitute the known values: P(Pencil | Pen) = P(Pen ∩ Pencil) / P(Pen) = 0.36 / 0.55.
  7. Perform the division: 0.36 ÷ 0.55 = 36/55.
  8. Simplify the fraction 36/55. Check for common factors: GCD of 36 and 55 is 1.
  9. Since 36 and 55 share no common factors other than 1, the fraction 36/55 is already in simplest form.
  10. Verify: 36/55 ≈ 0.6545, and 0.6545 × 0.55 = 0.36, confirming correctness.
  11. The conditional probability that an order contains a pencil given it contains a pen is 36/55.
  12. Final answer: 36/55.

Reference answer: 36/55

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