June 2024 Paper 3 Q18
18 The Human Resources director in a company is investigating the graduate status and salaries of its employees.
Event \(G\) is defined as the employee is a graduate.
Event \(H\) is defined as the employee earns at least £40 000 a year.
The director summarised the findings in the table of probabilities below.
| \(H\) | \(H^{\prime}\) | |
|---|---|---|
| \(G\) | 0.21 | 0.18 |
| \(G^{\prime}\) | 0.07 | 0.54 |
(a) An employee is selected at random.
(i) Find \(\mathrm{P}(G)\) [1 mark]
(ii) Find \(\mathrm{P}[(G \cap H)^{\prime}]\) [2 marks]
(iii) Find \(\mathrm{P}(H \mid G^{\prime})\) [2 marks]
(b) Determine whether the events \(G\) and \(H\) are independent.
Fully justify your answer. [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| (i) Obtains 0.39 | B1 | 1.1b |
| (1) | ||
| (ii) States or calculates \(1 - \mathrm{P}(G \cap H)\) or states 0.07 + 0.18 + 0.54 PI by correct answer | M1 | 1.1a |
| Obtains 0.79 | A1 | 1.1b |
| (2) | ||
| (iii) States \(\mathrm{P}(H \mid G^{\prime}) = \dfrac{\mathrm{P}(H \cap G^{\prime})}{\mathrm{P}(G^{\prime})}\) Condone missing \(\mathrm{P}(H \mid G^{\prime})\) or states \(\mathrm{P}(H \cap G^{\prime}) = 0.07\) or \(\dfrac{0.07}{k}\) seen or states 0.07 + 0.54 or 0.61 or \(\dfrac{k}{0.07 + 0.54}\) seen PI by correct answer | M1 | 1.1a |
| Obtains \(\dfrac{7}{61}\) or AWFW [0.11, 0.115] | A1 | 1.1b |
| (2) |
Typical solution
(i)
0.39
(ii)
\[1 - 0.21 = 0.79\](iii)
\[\mathrm{P}(H \mid G^{\prime}) = \frac{\mathrm{P}(H \cap G^{\prime})}{\mathrm{P}(G^{\prime})}\]\[= \frac{0.07}{0.61}\]\[= \frac{7}{61}\]| Scheme | Marks | AO |
|---|---|---|
| States their \(\mathrm{P}(G)\) from part 18(a)(i) × 0.28 or compares their \(\mathrm{P}(G)\) from part 18(a)(i) with 0.75 or compares their \(\mathrm{P}(H \mid G)\) with 0.28 or compares their \(\mathrm{P}(H \mid G^{\prime})\) from part 18(a)(iii) with 0.28 or any other valid comparison with one correct probability to at least 2 sf | M1 | 3.1b |
| Completes a reasoned argument and concludes that \(G\) and \(H\) are not independent | R1 | 2.4 |
| (2) | ||
| (7 marks) |
Typical solution
\[\mathrm{P}(G) \times \mathrm{P}(H) = 0.39 \times 0.28\]\[= 0.1092\]\[\mathrm{P}(G \cap H) = 0.21\]\[\mathrm{P}(G \cap H) \neq \mathrm{P}(G) \times \mathrm{P}(H)\]Hence \(G\) and \(H\) are not independent