# Question 2:

## Part 2(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Independently means arrival time of each car is unrelated to arrival time of any other car | E1 | Each answer must be 'in context' and 'clear'; allow sensible alternative wording |
| Randomly means arrival times of cars are not predictable | E1 | SC1 for ALL answers not in context but otherwise correct |
| At a uniform average rate means average rate of car arrivals does not vary over time | E1 | |
| **Total: 3** | | |

## Part 2(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{at most 1}) = e^{-0.62}\frac{0.62^0}{0!} + e^{-0.62}\frac{0.62^1}{1!}$ | M1 | For either term |
| $= 0.5379\ldots + 0.3335\ldots = 0.871$ | M1 | For sum of both |
| | A1 CAO | Allow 0.8715; not 0.872 or 0.8714; allow 0.87 without wrong working |
| **Total: 3** | | |

## Part 2(iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| New $\lambda = 10 \times 0.62 = 6.2$ | B1 | For mean (SOI) |
| $P(\text{more than 5 in 10 mins}) = 1 - 0.4141 = 0.5859$ | M1 | Use of $1 - P(X \leq 5)$ with any $\lambda$ |
| | A1 CAO | Allow 0.586 |
| **Total: 3** | | |

## Part 2(iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Poisson with mean 37.2 | B1 | |
| | B1 | Dependent on first B1; condone $P(37.2, 37.2)$ |
| **Total: 2** | | |

## Part 2(v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use Normal approximation with $\mu = \sigma^2 = \lambda = 37.2$ | B1 | For Normal (SOI) |
| | B1 | For parameters |
| $P(X \geq 40) = P\!\left(Z > \frac{39.5 - 37.2}{\sqrt{37.2}}\right)$ | B1 | For 39.5 |
| $= P(Z > 0.377) = 1 - \Phi(0.377) = 1 - 0.6469$ | M1 | Correct use of Normal approximation using correct tail |
| $= 0.3531$ | A1 cao | Allow 0.353 |
| **Total: 5** | | |

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