## Question 4(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Crystals must occur independently of one another | B1 | Allow interpreted, or "randomly" but nothing else. Must be contextualised; no other answers included. Ignore "singly". But allow "probability… is independent" |
| | **[1]** | |

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## Question 4(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^{-3.2}\dfrac{3.2^5}{5!} = \mathbf{0.114(0)}$ | M1 | Formula, or .0608 or .1781 or .1075 or .1203 (tables) |
| | A1 | Answer a.r.t. 0.114, implies both marks |
| | **[2]** | |

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## Question 4(iii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $Po(2.368)$ | M1 | $Po(0.74 \times 3.2)$ stated or implied. Allow for $0.75 \times 3.2$ etc, e.g. $Po(2.4)$ |
| $1 - e^{-2.368}\!\left(1 + 2.368 + \dfrac{2.368^2}{2}\right)$ | M1 | $1 -$ correct Poisson terms, their $\lambda$, allow $\pm 1$ term. Don't allow second M1 from $\lambda$ in tables, e.g. if MR, treat as E–1. |
| $= \mathbf{0.4219}$ | A1 | Answer, a.r.t. 0.422, implies all 3 marks. If no working: don't give M1A0 |
| | **[3]** | |

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## Question 4(iv):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $Po(32) \approx N(32, 32)$ | M1 | $N(\lambda, \lambda)$ stated or implied, allow $\sqrt{\lambda}$ or $\lambda^2$ for var. Needs $\lambda \geq 15$ |
| | A1 | $N(32, 32)$, allow $\sqrt{32}$ or $32^2$ for var |
| $1 - \Phi\!\left(\dfrac{35.5 - 32}{\sqrt{32}}\right)$ | M1 | Standardise with $\lambda$ and $\sqrt{\lambda}$ or $\lambda$, allow cc errors but not $\sqrt{n}$; both cc and $\sqrt{}$ correct. Can get (M0A0) M1A1 from $\lambda < 15$. Typically, no cc $\to 0.2203$, or $32 \to 0.4565$, 3/5 (but needs evidence, not just answer) |
| | A1 | both cc and $\sqrt{}$ correct |
| $= 1 - \Phi(.619) = \mathbf{0.2681}$ | A1 | Final answer, a.r.t. 0.268 |
| | **[5]** | |

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