NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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At a community library in Brisbane, patrons can borrow either a fiction book or a non-fiction book, and they can also choose to return the book by either the standard due date (S) or an extended due date (E). The library records show that 71 patrons borrowed a book, and 12 of them borrowed a fiction book with an extended return date.

Using a two-way table to represent all possible combinations of book type and return date, determine the exact probability that a randomly selected patron borrowed a fiction book AND chose the extended return date. 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

Worked solution and answer (click to reveal)

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Worked solution

  1. The problem states that 71 patrons borrowed a book, and 12 of them borrowed a fiction book with an extended return date. This is the number of patrons who satisfy both conditions: fiction book AND extended return date.
  2. The total number of patrons is 71. This is the sample space for the probability calculation.
  3. To find the probability of a randomly selected patron having borrowed a fiction book AND chosen the extended return date, use the formula: P(A and B) = number of favourable outcomes / total number of outcomes.
  4. Substitute the known values: P(fiction AND extended) = 12 / 71.
  5. Check if the fraction 12/71 can be simplified. The greatest common divisor (GCD) of 12 and 71 is 1, since 71 is a prime number and does not divide 12.
  6. Since the GCD is 1, the fraction 12/71 is already in its simplest form.
  7. The probability is therefore 12/71.
  8. Verify that the two-way table concept is correctly applied: the intersection of 'fiction' and 'extended' is 12, and the total is 71, so the probability is correctly calculated as 12/71.
  9. No further conversions or assumptions are needed, as all values are given and the fraction is exact.
  10. The final answer is 12/71, which is the exact probability.
  11. The result is a simplified fraction, and it matches the required format.
  12. The probability that a randomly selected patron borrowed a fiction book and chose the extended return date is exactly 12/71.

Reference answer: 12/71

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