June 2023 Paper 2 Q14
14 In this question you must show detailed reasoning.
A disease that affects trees shows no visible evidence for the first few years after the tree is infected.
A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not 100% reliable, and a researcher uses the following model.
- If the tree has the disease, the probability of a positive result is 0.95.
- If the tree does not have the disease, the probability of a positive result is 0.1.
Given that the result is positive, determine the probability that this tree has the disease. [3]
A forestry company wants to determine what proportion of trees in another county, \(B\), have the disease. They choose a large random sample of trees in county \(B\).
Each tree in the sample is tested and it is found that the result is positive for 43% of these trees.
| Scheme | Marks | AO |
|---|---|---|
| P(has disease | positive result) \(= \dfrac{\text{P(has disease \& positive result)}}{\text{P(positive result)}}\) | M1 | 3.4 |
| \(= \dfrac{0.35 \times 0.95}{0.35 \times 0.95 + 0.65 \times 0.1}\) | A1 | 1.1 |
| \(= 0.836\) (3 sf) | A1 | 1.1 |
| [3] |
Notes
M1: Attempting this calculation, allow wrong values but for this mark must be a fraction with a product in the numerator and a sum of two products in the denominator.
A1: Fully correct expression
A1: Or 133/159 or 0.8365 (4sf) (0.836477…)
| Scheme | Marks | AO |
|---|---|---|
| (Let proportion having the disease \(= p\)) \(p \times 0.95 + (1 - p) \times 0.1\) | M1 | 1.1 |
| \(p \times 0.95 + (1 - p) \times 0.1 = 0.43\) \(0.85p = 0.33\) | M1 | 3.4 |
| \(p = 0.388\) | A1 | 1.1 |
| About 39% of trees (in county \(B\)) have the disease | B1FT | 3.2a |
| [4] |
Notes
M1: Setting up an expression in this form using the given values
M1: Setting their expression = 0.43 and attempting to solve
A1: cao (watch for 0.389 from incorrect working)
B1FT: “Around 38.8 or 39 or 40” (oe e.g. 2/5).
Must be in context and include "about" or "approximately" or "roughly" oe