NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Compound-event simulations

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In a community garden in Adelaide, two separate irrigation systems are set to activate independently. System A activates with a probability of 72/100 on any given day, and System B activates with a probability of 74/100.

Using a digital simulation, the gardeners model 1000 independent daily activations. What is the exact theoretical probability that both systems activate on the same day?

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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Explanation

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Worked solution

  1. The problem states that System A activates with a probability of 72/100, and System B with 74/100. These are independent events, so the probability of both occurring is the product of their individual probabilities.
  2. Convert the given probabilities to decimal form for multiplication: 72/100 = 0.72, 74/100 = 0.74.
  3. Multiply the two probabilities: 0.72 × 0.74.
  4. Perform the multiplication: 0.72 × 0.74 = (72/100) × (74/100) = (72 × 74) / (100 × 100).
  5. Calculate 72 × 74 using the identity (a + b)(a - b) = a² - b², where a = 73, b = 1: 72 × 74 = (73 - 1)(73 + 1) = 73² - 1² = 5329 - 1 = 5328.
  6. Calculate the denominator: 100 × 100 = 10,000.
  7. So the combined probability is 5328 / 10,000.
  8. Simplify the fraction 5328 / 10,000 by finding the greatest common divisor (GCD) of 5328 and 10,000.
  9. Factor 5328: divide by 2 repeatedly: 5328 ÷ 2 = 2664, ÷2 = 1332, ÷2 = 666, ÷2 = 333. 333 ÷ 3 = 111, ÷3 = 37. So 5328 = 2⁴ × 3² × 37.
  10. Factor 10,000: 10,000 = 10⁴ = (2×5)⁴ = 2⁴ × 5⁴.
  11. The common factors are 2⁴ = 16. Divide numerator and denominator by 16: 5328 ÷ 16 = 333, 10,000 ÷ 16 = 625. So 5328 / 10,000 = 333 / 625.
  12. The simplified fraction 333/625 cannot be reduced further because 333 and 625 share no common factors (333 = 3²×37, 625 = 5⁴). Therefore, the exact theoretical probability that both systems activate on the same day is 333/625.

Reference answer: 333/625

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