# Question 4:

## Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Poisson | B1 | |

## Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Hits occur **singly** in time | B1 | Any one of the 3 statements — no context required. Must be constant (mean) rate, not constant probability or constant mean |
| Hits are **independent** or Hits occur **randomly** | B1 | A different statement with context of **hits**. NB random and independent are the same statement. If only one mark awarded, give 1st B1. Never award B0B1 |
| Hits occur at a **constant rate** | | |

## Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim Po(5)$ | B1 | Writing or using $Po(5)$ |
| $P(X=10) = P(X \leq 10) - P(X \leq 9)$ or $\frac{e^{-5}5^{10}}{10!}$ | M1 | |
| $= 0.9863 - 0.9682 = 0.0181$ | A1 | awrt 0.0181 |

## Part (d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim Po(10)$ | B1 | Writing or using $Po(10)$ |
| $P(X \geq 15) = 1 - P(X \leq 14)$ | M1 | Writing or using $1 - P(X \leq 14)$ |
| $= 1 - 0.9165 = 0.0835$ | A1 | awrt 0.0835 |

## Part (e):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim Po(50)$, Approximated by $N(50, 50)$ | B1B1 | 1st B1 for normal approximation. 2nd B1 for correct mean and sd |
| $P(X > 70) = P\!\left(Z > \frac{70.5 - 50}{\sqrt{50}}\right)$ | M1M1 | 1st M1 for attempting continuity correction $(71 \pm 0.5)$. 2nd M1 for standardising using their mean and sd; allow $[69.5, 70, 70.5, 71, 71.5]$, allow $\pm z$ |
| $= P(Z > 2.899\ldots)$ | A1 | $z = \pm$ awrt 2.9 or better |
| $= 1 - 0.9981$ | M1 | For $1 -$ tables value |
| $= 0.0019$ | A1 | awrt 0.0019 |
| SC using $P(X < 70.5/71.5) - P(X \leq 69.5/70.5)$ can get B1B1 M0M1A0 M0A0 | | |

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