# Question 2:

## Part (i)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = np = 12 \times 0.2 = 2.4$ | M1 | For product. If wrong $n$ used consistently throughout, allow M marks only |
| Do not allow subsequent rounding | A1 CAO | NB If they round to 2, even if they have obtained 2.4 first they get M1A0. For answer of '2.4 or 2 if rounded up' allow M1A0. Answer of 2 without working gets M0A0. If they attempt $E(X)$ by summing products $xp$ give no marks unless answer is fully correct |

**Total: 2 marks**

## Part (ii)(A)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $X \sim B(12, 0.2)$ | | |
| $P(\text{Wins exactly } 2) = \binom{12}{2} \times 0.2^2 \times 0.8^{10} = 0.2835$ | M1 | $0.2^2 \times 0.8^{10}$. With $p + q = 1$ |
| | M1 | $\binom{12}{2} \times p^2 q^{10}$. Also for $66 \times 0.004295$ |
| | A1 CAO | Allow answers within the range 0.283 to 0.284 with or without working or 0.28 to 0.283 if working shown |
| OR from tables: $0.5583 - 0.2749 = 0.2834$ | OR: M2 for $0.5583 - 0.2749$ | A1 CAO |

**Total: 3 marks**

## Part (ii)(B)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{Wins at least } 2) = 1 - P(X \leq 1) = 1 - 0.2749 = 0.7251$ | M1 | $P(X \leq 1)$. M1 0.2749 seen |
| | M1 | $1 - P(X \leq 1)$. M1 $1 - 0.2749$ seen |
| | A1 CAO | Allow 0.725 to 0.73 but not 0.72 |

**Point probability method:**
- $P(1) = 12 \times 0.2 \times 0.8^{11} = 0.2062$, $P(0) = 0.8^{12} = 0.0687$
- So $P(X \leq 1) = 0.2749$ gets M1 then mark as per scheme

**SC1** for $1 - P(X \leq 2) = 1 - 0.5583 = 0.4417$

For misread of tables value of 0.2749, allow 0 in (A) but MAX M1M1 in (B)

**Total: 3 marks**

## Part (iii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Let $p$ = probability that Ali wins a game | B1 | For definition of $p$ in context. Minimum needed for B1 is $p$ = probability that Ali wins. Allow $p$ = P(Ali wins) for B1. Definition of $p$ must include word probability (or chance or proportion or percentage but NOT possibility) |
| $H_0: p = 0.2$ | B1 | Allow $p=20\%$, allow $\theta$ or $\pi$ and $\rho$ but not $x$. However allow any single symbol if defined. Allow $H_0 = p=0.2$, Allow $H_0: p = \frac{2}{10}$. Do not allow $H_0: P(X=x) = 0.2$, $H_1: P(X=x) > 0.2$. Do not allow $H_0$: $=0.2$, $=20\%$, $P(0.2)$, $p(0.2)$, $p(x)=0.2$, $x=0.2$ (unless $x$ correctly defined as a probability). Do not allow $H_1: p \geq 0.2$. Do not allow $H_0$ and $H_1$ reversed for B marks but can still get E1 |
| $H_1: p > 0.2$ | B1 | Allow NH and AH in place of $H_0$ and $H_1$ |
| $H_1$ has this form as Ali claims that he is better at winning games than Mark is | E1 | For hypotheses given in words allow Maximum B0B1B1E1. Hypotheses in words must include probability (or chance or proportion or percentage) and the figure 0.2 oe |

**EITHER Probability method:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X \geq 7) = 1 - P(X \leq 6) = 1 - 0.9133 = 0.0867 > 5\%$ | B1 | For $P(X \geq 7)$. Do not award first B1 for poor symbolic notation such as $P(X=7) = 0.0867$ |
| | B1 | For 0.0867 Or $1 - 0.9133$ seen |
| | M1 | For comparison with 5% dep on B1 for 0.0867 |
| So not significant, so there is not enough evidence to reject the null hypothesis and we conclude that there is not enough evidence to suggest that Ali is better at winning games than Mark | A1 | For not significant or 'accept $H_0$' or 'cannot reject $H_0$' or 'reject $H_1$'. Must include 'not enough evidence' or something similar for E1 |
| | E1 dep on M1A1 | Zero for use of point prob $P(X=7) = 0.0546$ |

**OR Critical region method:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $X \sim B(20, 0.2)$ | | |
| $P(X \geq 7) = 1 - P(X \leq 6) = 1 - 0.9133 = 0.0867 > 5\%$ | B1 | For 0.0867. Allow any form of statement of CR e.g. $X \geq 8$, 8 to 20, 8 or above, $X > 8$, $\{8, \ldots\}$, annotated number line, etc but not $P(X \geq 8)$ |
| $P(X \geq 8) = 1 - P(X \leq 7) = 1 - 0.9679 = 0.0321 < 5\%$ | B1 | For 0.0321. $\{8,9,10,11,12\}$ gets max B2M1A0 – tables stop at 8 |
| So critical region is $\{8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}$ | M1 | For at least one comparison with 5%. NB USE OF POINT PROBABILITIES gets B0B0M0A0 |
| 7 does not lie in the critical region, so not significant | A1 CAO | For critical region and not significant or 'accept $H_0$' or 'cannot reject $H_0$' or 'reject $H_1$' dep on M1 and at least one B1. Providing there is sight of 95%, allow B1 for 0.9133, B1 for 0.9679, M1 for comparison with 95% A1CAO for correct CR |
| So there is not enough evidence to reject the null hypothesis and we conclude that there is not enough evidence to suggest that Ali is better at winning games than Mark | E1 dep on M1A1 | See additional notes below the scheme for other possibilities. **PLEASE CHECK THAT THERE IS NO EXTRA WORKING ON THE SECOND PAGE IN THE ANSWER BOOKLET** |

**Total: 5 marks**

**TOTAL: 17 marks**