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

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A craft designer in Melbourne creates a spinner divided into 18 equal sectors, numbered 1 to 18. The designer conducts a two-step chance experiment: spinning the spinner twice and recording the sum of the two outcomes.

What is the exact probability that the sum is exactly 18? 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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Explanation

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Worked solution

  1. The spinner has 18 equal sectors numbered 1 to 18, so each spin has 18 possible outcomes, each with equal probability.
  2. The experiment consists of two independent spins, so the total number of possible outcomes is 18 × 18 = 324.
  3. We are interested in the sum of the two spins being exactly 18. We need to count how many pairs (a, b) satisfy a + b = 18, where a and b are integers from 1 to 18.
  4. List all valid pairs (a, b) such that a + b = 18 and 1 ≤ a ≤ 18, 1 ≤ b ≤ 18. Start with a = 1: b = 17 → (1,17). a = 2: b = 16 → (2,16). Continue this pattern.
  5. The pairs are: (1,17), (2,16), (3,15), (4,14), (5,13), (6,12), (7,11), (8,10), (9,9), (10,8), (11,7), (12,6), (13,5), (14,4), (15,3), (16,2), (17,1).
  6. Count the number of such pairs: there are 17 pairs.
  7. Each outcome (a,b) is equally likely, so the probability of sum = 18 is the number of favourable outcomes divided by total outcomes: 17 / 324.
  8. Check if 17/324 can be simplified. The greatest common divisor of 17 and 324 is 1, since 17 is prime and does not divide 324.
  9. Therefore, the fraction 17/324 is already in simplest form.
  10. Verify by checking the minimum and maximum possible sums: minimum sum is 1+1=2, maximum is 18+18=36. 18 is within range, and we have listed all valid pairs.
  11. The probability that the sum is exactly 18 is 17/324.
  12. Final answer: 17/324.

Reference answer: 17/324

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