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

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A digital simulation in Brisbane models rolling two standard eight-sided dice (numbered 1 to 8) independently. The compound event of interest is that the absolute difference between the two numbers rolled is exactly 6.

Using theoretical probability, what is the exact probability of this event occurring in a single trial? 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.
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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Worked solution

  1. Two standard eight-sided dice are rolled independently. Each die has faces numbered from 1 to 8. The total number of possible outcomes is 8 × 8 = 64.
  2. The compound event of interest is that the absolute difference between the two dice is exactly 6. That is, |A - B| = 6, where A and B are the values rolled on the first and second die.
  3. List all pairs (A, B) such that |A - B| = 6. This occurs when A = B + 6 or B = A + 6.
  4. Find all valid (A, B) pairs where A and B are integers from 1 to 8 and |A - B| = 6. Start with A = 1: B = 7 (|1 - 7| = 6) → (1,7).
  5. A = 2: B = 8 → (2,8). A = 3: B = 9 (invalid). So only (1,7) and (2,8) for A < B.
  6. Now consider B = A + 6: A = 7 → B = 1 → (7,1). A = 8 → B = 2 → (8,2).
  7. List all valid pairs: (1,7), (2,8), (7,1), (8,2).
  8. Count the number of favourable outcomes: 4.
  9. Total number of possible outcomes: 64.
  10. The theoretical probability is the number of favourable outcomes divided by total outcomes: 4 / 64.
  11. Simplify the fraction: 4/64 = 1/16.
  12. The exact probability of the absolute difference being exactly 6 is 1/16.

Reference answer: 1/16

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