NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Relative frequency and event combinations

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In a school materials order, 70 students received at least one of two items: a geometry set or a scientific calculator. 68 students received a geometry set, and 42 students received both items. Using a two-way table, calculate the exact relative frequency of students who received a geometry set or a scientific calculator (inclusive or), based on the total number of students who received at least one item.

Express your answer as a simplified fraction.

Concept illustrations

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An event and its complement fill the whole. The first 3 of 10 equal parts represent rain, probability 0.3. The other 7 represent no rain, probability 0.7. Together they sum to 1.Rain: 0.3No rain: 0.70.3 + 0.7 = 1
The first 3 of 10 equal parts represent rain, probability 0.3. The other 7 represent no rain, probability 0.7. Together they sum to 1.
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.

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Worked solution

  1. Given: 70 students received at least one item (geometry set or scientific calculator). 68 received a geometry set. 42 received both items. We need to find the relative frequency of students who received a geometry set or a scientific calculator (inclusive or), based on the total number of students who received at least one item.
  2. Let G be the set of students who received a geometry set. Let C be the set of students who received a scientific calculator. We are given |G ∪ C| = 70 (students who received at least one), |G| = 68, and |G ∩ C| = 42.
  3. We use the formula for the union of two sets: |G ∪ C| = |G| + |C| - |G ∩ C|. We know |G ∪ C| = 70, |G| = 68, |G ∩ C| = 42. We solve for |C|.
  4. Substitute known values: 70 = 68 + |C| - 42.
  5. Simplify: 70 = 26 + |C|.
  6. Solve for |C|: |C| = 70 - 26 = 44.
  7. Now, the number of students who received a geometry set or a scientific calculator (inclusive or) is |G ∪ C| = 70.
  8. The relative frequency is the number of students who received at least one item divided by the total number of students who received at least one item. So, relative frequency = |G ∪ C| / |G ∪ C| = 70 / 70.
  9. Simplify: 70 / 70 = 1.
  10. The relative frequency is 1, which as a simplified fraction is 1/1.
  11. Check: 68 received geometry set, 44 received calculator, 42 received both. So, only 68 + 44 - 42 = 70 received at least one. This matches the given total. The inclusive or includes all 70, so the relative frequency is 1.
  12. Therefore, the relative frequency of students who received a geometry set or a scientific calculator (inclusive or) is 1.

Reference answer: 1

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