S2 June 2005 Q3
3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.
(a) State two conditions under which a Poisson distribution is a suitable model for \(X\). (2)
The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
(b) a randomly chosen page has no misprints, (2)
(c) the total number of misprints on 2 randomly chosen pages is more than 7. (3)
The first chapter contains 20 pages.
(d) Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints. (7)
| Scheme | Marks |
|---|---|
| Misprints are random / independent, occur singly in space and at a constant rate | B1, B1 |
| (2) |
Notes
B1, B1 context, any 2
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X = 0) = \mathrm{e}^{-2.5}\) | M1 |
| \(= 0.08208\ldots = 0.0821\) | A1 |
| (2) |
Notes
M1 Po(2.5)
A1 0.0821
| Scheme | Marks |
|---|---|
| \(Y \sim \mathrm{Po}(5)\) for 2 pages | B1 |
| \(\mathrm{P}(Y \gt 7) = 1 - \mathrm{P}(X \leqslant 7)\) | M1 |
| \(= 1 - 0.8666 = 0.1334\) | A1 |
| (3) |
Notes
B1 implied
M1 use of 1 – and correct inequality
A1 0.1334
| Scheme | Marks |
|---|---|
| For 20 pages, \(Y \sim \mathrm{Po}(50)\) | B1 |
| \(Y \sim \mathrm{N}(50, 50)\) approx | B1 |
| \(\mathrm{P}(Y \lt 40) = \mathrm{P}(Y \leqslant 39.5)\) | M1 |
| \(= \mathrm{P}\left(Z \leqslant \dfrac{39.5 - 50}{\sqrt{50}}\right)\) | M1 A1 |
| \(= \mathrm{P}(Z \leqslant -1.4849)\) | A1 |
| \(= 1 - 0.93 = 0.07\) | A1 |
| (7) | |
| (14 marks) |
Notes
1st B1 Po(50)
2nd B1 N(50, 50)
1st M1 cc \(\pm 0.5\)
2nd M1 standardise above
1st A1 all correct
2nd A1 awrt \(-1.48\)
3rd A1 0.07