NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Probability experiments and simulations

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A probability simulation is conducted in Melbourne using a digital tool to model a two-step experiment. A bag contains 23 red marbles and 64 blue marbles.

Two marbles are drawn consecutively without replacement. What is the exact probability that both marbles are red?

Express your answer as a simplified fraction.

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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 bag contains 23 red marbles and 64 blue marbles, so the total number of marbles is 23 + 64 = 87.
  2. The first marble is drawn without replacement, so the probability of drawing a red marble on the first draw is 23/87.
  3. After one red marble is drawn, there are now 22 red marbles left and 86 marbles in total.
  4. The probability of drawing a red marble on the second draw, given that the first was red, is 22/86.
  5. Since the two events are dependent (without replacement), the combined probability is the product of the individual probabilities: (23/87) × (22/86).
  6. Multiply the numerators: 23 × 22 = 506.
  7. Multiply the denominators: 87 × 86 = 7482.
  8. The combined probability is 506/7482.
  9. Simplify the fraction by finding the greatest common divisor (GCD) of 506 and 7482. First, factor 506: 506 = 2 × 11 × 23. Factor 7482: 7482 = 2 × 3 × 13 × 97. The common factor is 2.
  10. Divide both numerator and denominator by 2: 506 ÷ 2 = 253, 7482 ÷ 2 = 3741. The simplified fraction is 253/3741.
  11. Check if 253 and 3741 have any common factors. 253 = 11 × 23. 3741 ÷ 11 = 340.09 (not integer), 3741 ÷ 23 = 162.65 (not integer). No further common factors exist.
  12. Therefore, the exact probability that both marbles drawn are red is 253/3741.

Reference answer: 253/3741

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