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

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In a community garden in Brisbane, 25 plots are allocated for vegetable growing and 42 plots are used for flower cultivation. Of the 25 vegetable plots, 18 also include a small water feature.

If a plot is selected at random and is known to have a water feature, what is the probability that it is used for vegetables? Express your answer as an exact fraction.

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Restrict the sample space before counting. All 12 dots are cyclists. Three filled dots also represent red clothing. Within the cyclist group, P(red given cycles) = 3/12 = 1/4.12 cyclists · 3 wearing red
All 12 dots are cyclists. Three filled dots also represent red clothing. Within the cyclist group, P(red given cycles) = 3/12 = 1/4.
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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. Identify the total number of plots with a water feature. From the scenario, 18 vegetable plots include a water feature, and no information is given about water features in flower plots. Therefore, only the 18 vegetable plots with water features are counted.
  2. Determine the total number of plots that have a water feature. Since only vegetable plots are stated to have water features, and 18 such plots exist, the total number of plots with a water feature is 18.
  3. Define the conditional probability required: the probability that a plot is used for vegetables given that it has a water feature. This is written as P(vegetable | water feature).
  4. Apply the definition of conditional probability: P(A|B) = P(A and B) / P(B). Here, A is 'vegetable plot', B is 'has water feature'. So P(vegetable | water feature) = P(vegetable and water feature) / P(water feature).
  5. Calculate P(vegetable and water feature): this is the proportion of plots that are both vegetable plots and have a water feature. There are 18 such plots out of a total of 25 + 42 = 67 plots. So P(vegetable and water feature) = 18 / 67.
  6. Calculate P(water feature): this is the proportion of plots that have a water feature. Since only vegetable plots have water features, and 18 such plots exist, P(water feature) = 18 / 67.
  7. Substitute into the conditional probability formula: P(vegetable | water feature) = (18 / 67) / (18 / 67).
  8. Simplify the expression: (18 / 67) ÷ (18 / 67) = 1. This would be correct if both numerator and denominator were the same, but we must re-evaluate the logic.
  9. Re-evaluate: P(vegetable and water feature) = 18 / 67, and P(water feature) = 18 / 67, so P(vegetable | water feature) = (18 / 67) / (18 / 67) = 1. But this implies that all plots with water features are vegetable plots, which is true by the given data.
  10. Since all 18 plots with water features are vegetable plots, and no flower plots are said to have water features, the probability that a plot with a water feature is used for vegetables is 1.
  11. Express the result as an exact fraction: 1 = 1/1.
  12. Final answer: given that a plot has a water feature, it must be a vegetable plot, so the probability is 1/1.

Reference answer: 1/1

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