NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Compound-event simulations

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A student in Melbourne uses a digital simulation to model rolling two custom eight-sided dice, each numbered from 1 to 8. The compound event of interest is rolling a difference of 5 between the two dice.

Using theoretical probability, what is the exact probability of this event occurring in a single trial? Express your answer as a simplified fraction.

Concept illustrations

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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.
Six equally likely die outcomes. Outcomes 5 and 6 are greater than 4: 2 favourable outcomes out of 6, so probability = 1/3.12345✓6✓Each outcome has probability 1/6
Outcomes 5 and 6 are greater than 4: 2 favourable outcomes out of 6, so probability = 1/3.

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Explanation

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Worked solution

  1. The experiment involves rolling two eight-sided dice, each numbered from 1 to 8. The total number of possible outcomes is calculated by multiplying the number of outcomes for each die: 8 × 8 = 64.
  2. The compound event of interest is a difference of 5 between the two dice. This means |die1 - die2| = 5.
  3. List all pairs (die1, die2) where the absolute difference is 5. Start with die1 = 1: |1 - die2| = 5 → die2 = 6. So (1,6) is valid.
  4. Continue for die1 = 2: |2 - die2| = 5 → die2 = 7. So (2,7) is valid.
  5. die1 = 3: |3 - die2| = 5 → die2 = 8. So (3,8) is valid.
  6. die1 = 4: |4 - die2| = 5 → die2 = 9 or -1. But die2 must be between 1 and 8. No valid pair.
  7. die1 = 5: |5 - die2| = 5 → die2 = 10 or 0. Invalid. No valid pair.
  8. die1 = 6: |6 - die2| = 5 → die2 = 1 or 11. die2 = 1 is valid. So (6,1) is valid.
  9. die1 = 7: |7 - die2| = 5 → die2 = 2 or 12. die2 = 2 is valid. So (7,2) is valid.
  10. die1 = 8: |8 - die2| = 5 → die2 = 3 or 13. die2 = 3 is valid. So (8,3) is valid.
  11. The valid pairs are: (1,6), (2,7), (3,8), (6,1), (7,2), (8,3). Total of 6 favourable outcomes.
  12. The probability is the number of favourable outcomes divided by total outcomes: 6 / 64 = 3 / 32. This is the simplified fraction.

Reference answer: 3/32

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