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

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In Melbourne, a survey of 1200 residents found that 720 people own a car and 480 people have a valid driver's licence. Of those who own a car, 600 also have a driver's licence.

What is the probability that a randomly selected person who owns a car also has a driver's licence? Express your answer as an exact fraction.

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Worked solution

  1. The question asks for the probability that a randomly selected person who owns a car also has a driver's licence. This is a conditional probability: P(has licence | owns car).
  2. From the survey data: 720 people own a car, and of those, 600 have a valid driver's licence.
  3. The conditional probability is calculated as: P(B|A) = P(A and B) / P(A). Here, A is 'owns a car', B is 'has a driver’s licence'.
  4. The number of people who both own a car and have a driver’s licence is 600.
  5. The number of people who own a car is 720.
  6. Substitute into the formula: P(has licence | owns car) = 600 / 720.
  7. Simplify the fraction 600/720 by dividing both numerator and denominator by their greatest common divisor. First, divide both by 10: 60/72.
  8. Divide both 60 and 72 by 12: 60 ÷ 12 = 5, 72 ÷ 12 = 6. So, 600/720 = 5/6.
  9. The simplified fraction is 5/6. This is the exact probability.
  10. Check: 5/6 of 720 = (5 × 720) / 6 = 3600 / 6 = 600. This matches the given number of people who own a car and have a licence.
  11. The probability that a randomly selected car owner also has a driver’s licence is exactly 5/6.
  12. Final answer is 5/6, expressed as a simplified fraction.

Reference answer: 5/6

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