NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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At a sports equipment distribution event in Sydney, 95 participants are choosing between two types of sports gear: a football (F) or a netball (N), and two types of protective gear: shin guards (S) or knee pads (K). The event organisers record the choices using a two-way table.

It is known that 84 participants chose football, 45 chose shin guards, and 30 chose both football and shin guards. Using a two-way table, determine the exact probability that a randomly selected participant chose netball AND knee pads.

Express your answer as a simplified fraction.

Concept illustrations

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Two independent fair coin tosses. HH, HT, TH and TT are equally likely. Two outcomes have exactly one head: probability = 2/4 = 1/2.HHHT✓TH✓TTExactly one head: HT or TH
HH, HT, TH and TT are equally likely. Two outcomes have exactly one head: probability = 2/4 = 1/2.
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. Given: Total participants = 95, Football (F) = 84, Shin guards (S) = 45, F and S = 30. Need to find P(netball AND knee pads).
  2. Construct a two-way table with rows: Football (F), Netball (N); columns: Shin guards (S), Knee pads (K).
  3. Fill in known values: F ∩ S = 30. F total = 84, so F ∩ K = 84 - 30 = 54.
  4. S total = 45, so N ∩ S = 45 - 30 = 15.
  5. N total = 95 - 84 = 11. So N ∩ K = 11 - 15 = -4? Wait: N ∩ S = 15, so N ∩ K = 11 - 15 = -4? Invalid. Recalculate: N total = 95 - 84 = 11. N ∩ S = 15, but 15 > 11, impossible.
  6. Check: S total = 45, F ∩ S = 30, so N ∩ S = 45 - 30 = 15. N total = 95 - 84 = 11. But N ∩ S = 15 > 11, impossible. Contradiction. Recheck given data.
  7. Recheck: Total = 95, F = 84, so N = 11. S = 45, F ∩ S = 30. So N ∩ S = 45 - 30 = 15. But N = 11, so N ∩ S cannot exceed 11. 15 > 11 → contradiction. Therefore, data is inconsistent.
  8. Re-express: F ∩ S = 30, F = 84, so F ∩ K = 54. S = 45, so N ∩ S = 15. But N = 11, so N ∩ S = 15 > 11 → impossible. Therefore, the given values are inconsistent with total participants.
  9. Verify sum: F ∩ S = 30, F ∩ K = 54, N ∩ S = 15, N ∩ K = ? Total = 30 + 54 + 15 + N ∩ K = 99 + N ∩ K. But total must be 95. So 99 + N ∩ K = 95 → N ∩ K = -4. Impossible.
  10. Conclusion: The given data (F=84, S=45, F∩S=30, total=95) leads to a negative count for N ∩ K, which is impossible. Therefore, the scenario is mathematically inconsistent.
  11. No valid solution exists under given constraints. The data contradicts basic counting principles.
  12. The problem as stated contains inconsistent data and cannot yield a valid probability.

Reference answer: Error: Inconsistent data

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