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

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A digital simulation in Adelaide models a two-step chance experiment using a bag containing 67 red balls and 79 blue balls. Two balls are drawn consecutively without replacement.

What is the exact probability that the first ball is red and the second ball is blue? 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 67 red balls and 79 blue balls, so the total number of balls is 67 + 79 = 146.
  2. The first ball is drawn without replacement, so the probability that it is red is the number of red balls divided by the total number of balls: 67/146.
  3. After drawing one red ball, there are now 66 red balls and 79 blue balls left, so the total number of balls is 145.
  4. Given that the first ball was red, the probability that the second ball is blue is 79/145.
  5. The combined probability of both events occurring (first red, then blue) is the product of the individual probabilities: (67/146) × (79/145).
  6. Multiply the numerators: 67 × 79 = 5293.
  7. Multiply the denominators: 146 × 145 = 21170.
  8. The combined probability is 5293/21170.
  9. Check for common factors between 5293 and 21170. First, test divisibility by small primes.
  10. Check if 5293 and 21170 share any common prime factors. 5293 is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, or 71. 21170 is divisible by 2, 5, and 10. 5293 is odd and not divisible by 5, so no common factor of 2 or 5.
  11. Since 5293 and 21170 have no common prime factors, the fraction 5293/21170 is already in simplest form.
  12. Therefore, the exact probability that the first ball is red and the second is blue is 5293/21170.

Reference answer: 5293/21170

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