**(i)** W = wrong, C = correct

Two different tree diagrams shown with branching:

First option (3 branches first qn and 2 by 2 for second qn only):
- 3 branches first qn and 2 by 2 for second qn only | M1

- One branch twice for third qn or two branches twice with 0 and 1 seen on branches | M1

- Any two of $\frac{1}{3}$, $\frac{1}{3}$ and 1 seen as probs | B1

- Probs all correct and sensible labels NB SR for 4 outcomes instead of 3, M1 B1 only | A1 4

OR

- 2 branches first qn and 1 by 2 for second qn only | M1

- One branch once for third qn or two branches with 0 and 1 seen on branches | M1

- Any two of $\frac{2}{3}$ or $\frac{2}{3}$, $\frac{1}{2}$ and 1 seen as probs or Probs all correct and sensible labels | B1, A1

**(ii)**

| $x$ | 1 | 2 | 3 |
|-----|-------|-------|-------|
| Prob | $\frac{1}{3}$ | $\frac{1}{3}$ | $\frac{1}{3}$ |

| B1 | $1, 2, 3$ seen only or

| B1 | 2 correct probs

| B1 | 3 correct probs

$$P(1) = P(C) \text{ say } = \frac{1}{3}$$

$$P(2) = P(WC) = \frac{1}{6} \quad P(WC) = \frac{1}{6} \text{ total } P(2) = \frac{1}{3}$$

$$P(3) = P(WWC) = \frac{1}{6} \quad P(WWC) = \frac{1}{6} \text{ total } P(3) = \frac{1}{3}$$

$$E(X) = 1 \times \frac{1}{3} + 2 \times \frac{1}{3} + 3 \times \frac{1}{3} = 2$$ | B1 $\checkmark$ 4 | Correct answer iff their probs provided $0.999 \leq \sum p \leq 1$

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