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

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A sports equipment manager in Brisbane uses a digital simulation to allocate two balls from a pool of 70 footballs and 84 basketballs, selected without replacement. The simulation models the probability of selecting one football followed by one basketball.

What is the exact probability of this outcome? 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 total number of balls is 70 footballs + 84 basketballs = 154 balls.
  2. The first selection is a football. The probability of selecting a football first is 70/154.
  3. After selecting one football, there are now 153 balls left, with 84 basketballs still available.
  4. The probability of selecting a basketball second, given a football was selected first, is 84/153.
  5. The combined probability of selecting a football followed by a basketball (without replacement) is the product of the two probabilities: (70/154) × (84/153).
  6. Multiply the numerators: 70 × 84 = 5880.
  7. Multiply the denominators: 154 × 153 = 23562.
  8. The fraction is 5880/23562.
  9. Simplify the fraction by finding the greatest common divisor (GCD) of 5880 and 23562. First, factor both numbers.
  10. Factor 5880: 5880 = 2^3 × 3 × 5 × 7^2. Factor 23562: 23562 = 2 × 3^2 × 13 × 101. The common factors are 2 and 3, so GCD = 2 × 3 = 6.
  11. Divide numerator and denominator by 6: 5880 ÷ 6 = 980, 23562 ÷ 6 = 3927. The simplified fraction is 980/3927.
  12. Check if 980 and 3927 have any common factors. 980 = 2^2 × 5 × 7^2; 3927 = 3 × 7 × 11 × 17. The only common factor is 7. 980 ÷ 7 = 140, 3927 ÷ 7 = 561. New fraction: 140/561. 140 and 561 share no further common factors. Final simplified fraction is 140/561.

Reference answer: 140/561

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