S2 June 2005 Q5
5. In a manufacturing process, 2% of the articles produced are defective. A batch of 200 articles is selected.
(a) Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles. (5)
(b) Estimate the probability there are less than 5 defective articles. (2)
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{B}(200, 0.02)\) | B1 |
| \(n\) large, \(p\) small so \(X \sim \mathrm{Po}(np) = \mathrm{Po}(4)\) | B1, B1 |
| \(\mathrm{P}(X = 5) = \dfrac{\mathrm{e}^{-4}4^5}{5!}\) | M1 |
| \(= 0.1563\) | A1 |
| (5) |
Notes
1st B1 implied
B1, B1 conditions, Po(4)
M1 \(\mathrm{P}(X \leqslant 5) - \mathrm{P}(X \leqslant 4)\)
A1 0.1563
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \lt 5) = \mathrm{P}(X \leqslant 4)\) | M1 |
| \(= 0.6288\) | A1 |
| (2) | |
| (7 marks) |
Notes
M1 \(\mathrm{P}(X \leqslant 4)\)
A1 0.6288