NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Conditional probability

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In a study of 1800 residents in Perth, 1080 people have a home internet subscription, and 720 people have a streaming service subscription. Of those who have a home internet subscription, 540 also have a streaming service subscription.

If a resident is selected at random and is known to have a streaming service subscription, what is the probability that they also have a home internet subscription? Express your answer as an exact fraction.

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Explanation

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Worked solution

  1. Given: 1800 residents in Perth. 1080 have a home internet subscription (I). 720 have a streaming service subscription (S). 540 have both (I ∩ S).
  2. We are asked to find the probability that a resident has a home internet subscription given they have a streaming service subscription. This is a conditional probability: P(I | S).
  3. The formula for conditional probability is P(A | B) = P(A ∩ B) / P(B). Here, A is having internet (I), B is having streaming (S).
  4. P(I ∩ S) = number of people with both / total = 540 / 1800.
  5. P(S) = number with streaming / total = 720 / 1800.
  6. Substitute into formula: P(I | S) = (540 / 1800) / (720 / 1800).
  7. The 1800 in denominator cancels out: (540 / 1800) ÷ (720 / 1800) = 540 / 720.
  8. Simplify 540 / 720. Divide numerator and denominator by 180: 540 ÷ 180 = 3, 720 ÷ 180 = 4.
  9. Simplify further: 3 / 4.
  10. Check: 540 / 720 = 0.75, and 3/4 = 0.75. Correct.
  11. The probability that a resident with a streaming service subscription also has a home internet subscription is 3/4.
  12. Final answer is 3/4.

Reference answer: 3/4

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