S1 January 2005 Q5
5. Articles made on a lathe are subject to three kinds of defect, \(A\), \(B\) or \(C\). A sample of 1000 articles was inspected and the following results were obtained.
31 had a type \(A\) defect
37 had a type \(B\) defect
42 had a type \(C\) defect
11 had both type \(A\) and type \(B\) defects
13 had both type \(B\) and type \(C\) defects
10 had both type \(A\) and type \(C\) defects
6 had all three types of defect.
(a) Draw a Venn diagram to represent these data. (6)
Find the probability that a randomly selected article from this sample had
(b) no defects, (1)
(c) no more than one of these defects. (2)
An article selected at random from this sample had only one defect.
(d) Find the probability that it was a type \(B\) defect. (2)
Two different articles were selected at random from this sample.
(e) Find the probability that both had type \(B\) defects. (2)
| Scheme | Marks |
|---|---|
| 6 | B1 |
| subtract | M1 |
| 4, 5, 7 | A1 |
| subtract | A1 |
| 16, 19, 25 | A1 |
| 918 | B1 |
| (6) |
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{No defects}) = \dfrac{918}{1000} = 0.918\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{No more than 1}) = \dfrac{918 + 16 + 19 + 25}{1000}\) OR \(1 - \dfrac{5 + 6 + 4 + 7}{1000}\) | M1 |
| \(= 0.978\) | A1 |
| (2) |
Notes
A1 0.978
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(B \mid \text{Only 1 defect}) = \dfrac{\mathrm{P}(B \text{ and 1 defect})}{\mathrm{P}(\text{1 defect})} = \dfrac{\frac{19}{1000}}{\frac{16 + 19 + 25}{1000}}\) | M1 |
| \(= \dfrac{19}{60}\) | A1 |
| (2) |
Notes
M1 conditional prob
A1 \(\dfrac{19}{60}\) or \(0.31\dot{6}\) or 0.317
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{Both had type } B) = \dfrac{37}{1000} \times \dfrac{36}{999}\) | M1 |
| \(= \dfrac{37}{27750}\) or \(0.001\dot{3}\) or 0.00133 | A1 |
| (2) | |
| (13 marks) |
Notes
M1 theirs from \(B\) ×
A1 cao