| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(X) = -2p - p + 0 + \frac{1}{2} + 3p \quad ; \quad = \frac{1}{2}$ | M1 ; A1 | For a correct expr'n for E(X) in p (at least 3 non-zero terms seen). May be implied by A1. For $\frac{1}{2}$ (or exact equivalent e.g. $\frac{4}{8}$ or 0.5) |
| $E(X^2) = 4p + p + 0 + 1 + 9p = [14p + 1]$ $[\text{Var}(X)] = E(X^2) - [E(X)]^2 = 14p + 1 - (\text{"}_{2}\text{"})^2$ | M1A1, dM1 | For a correct expression for E(X²) (at least 3 non-zero terms). May be implied by A1. For $14p + 1$ or any fully correct expression in terms of p. For use of $[\text{Var}(X)] = E(X^2) - [E(X)]^2$. If they think E(X²) = Var(X) get $p = \frac{3}{8}, q = \frac{3}{8}$ and up to (b) M1A1M0M1A0 (c) B1ft and if they get $\frac{319}{384}$ in (d) it implies M1M1A1A0 there and access to (e) |
| So $14p + 0.75 = 2.5$ | M1 | For forming a linear equation in p using the 2.5 |
| $p = \frac{1}{8}$ | A1 | For $p = \frac{1}{8}$ or exact equivalent e.g. 0.125 |
| Sum of probabilities = 1 implies $q = \frac{3}{8}$ | B1ft | For $q = \frac{3}{8}$ or exact equivalent e.g. 0.375 or $\frac{3}{8} - 3p$ 0<p<1 |
| $P(\text{Amar wins}) = \text{e.g. } P(X_1 > 0) + P(X_1 < 0) \times P([\text{X_1 + X_2] > 0 | X_1 < 0})$ **or** $P(X_1 = 2 \text{ or } 3) + P(X_1 = -2) \times P(X_2 = 3) + P(X_1 = -1) \times P(X_2 = 2 \text{ or } 3)$ | M1, M1 | For identifying only the correct cases (any correct list, adding not needed). For identifying all the cases where a 2nd spin is required and probabilities (no extras) |
| **Cases** $X_1 = -2$ and $X_2 = 3$ so probability = $p^2$ $X_1 = -1$ and $X_2 \geq 2$ so probability = $p(p + \frac{1}{4})$ | | |
| Total probability = $p + 0.25 + p^2 + p(p + 0.25) = \frac{1}{8} + \frac{1}{4} + \frac{1}{64} + \frac{1}{64} + \frac{1}{32}$ | A1ft | For correct expression for total probability (allow their $0 < p < 1$ letter p) |
| $= \frac{7}{16}$ | A1 | |
| $[P(\text{win}) < 0.5$ **Amar should not play the game or "disagree"** | M1, A1 | For identifying that the important feature is that P(win) < 0.5 (o.e.) [ft their $\frac{7}{16} < 0.5$]. For concluding that he shouldn't play the game (dep on M1 seen & 0.375 < (d) < 0.5) |

**Notes:**

**(a)** M1 for a correct expr'n for E(X) in p (at least 3 non-zero terms seen). May be implied by A1
A1 for $\frac{1}{2}$ (or exact equivalent e.g. $\frac{4}{8}$ or 0.5)

**(b)** 1st M1 for a correct expression for E(X²) (at least 3 non-zero terms). May be implied by A1
1st A1 for $14p + 1$ or any fully correct expression in terms of p
2nd dM1 dep on 1st M1 for use of $[\text{Var}(X)] = E(X^2) - [E(X)]^2$
3rd M1 for forming a linear equation in p using the 2.5
2nd A1 for $p = \frac{1}{8}$ or exact equivalent e.g. 0.125

**(c)** B1ft for $q = \frac{3}{8}$ or exact equivalent e.g. 0.375 or $\frac{3}{8} - 3p$ 0<p<1

**(d)** 1st M1 for identifying only the correct cases (any correct list, adding not needed)
2nd M1 for identifying all the cases where a 2nd spin is required and probabilities (no extras)
1st A1ft for correct expression for total probability (allow their $0 < p < 1$ letter p)
2nd A1 for $\frac{7}{16}$ (or exact equivalent e.g. 0.4375) [$\frac{7}{16}$ with no incorrect working seen gets 4/4]

**ALT** Allow P(loses) = q + p(1 – p) + p · (0.75 – p) only if 1 – P(loses) is seen

**(e)** M1 for identifying that the important feature is that P(win) < 0.5 (o.e.) [ft their $\frac{7}{16} < 0.5$]
A1cao for concluding that he shouldn't play the game (dep on M1 seen & 0.375 < (d) < 0.5)

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