| Consistent use of $\frac{1}{3}$ or MR of 30% (eg 0.2): (i) B1B0B1B1 (iia) B0 (iib) 0.7901–0.4609 or $^6C_3(\frac{2}{3})^3(\frac{1}{3})^3$ M1; = 0.329 (3 sf) A1 (iii) $p$ = "0.3292" M1; $^7C_3(1–"0.3292")^4(0.3292)^3$ M1; = 0.253 (3 sf) A1 | ie max 8/10 | | Allow mis-spellings but NOT "Biometric". Condone B–(5, 0.3) or B(0.3, 5); B1B1 but B(X = 0.3, $n$ = 5): B1B0 |
| Prob of gift same for all pkts | B1 | Prob of gift is constant or fixed or consistent or same oe | NOT: prob of success const; NOT prob stays same each go |
| Whether pkt contains gift is indep of other pkts | B1 4 | Obtaining a gift is indep. Each time receive a gift is indep. Context needed for 3rd & 4th B-mks | One box doesn't affect another. Pkts indep. Gifts indep. She buys packets separately. Prob of a gift is indep. Prob of gift indep of one another & const: B1B1. NOT: Each week is indep. NOT: Number of gifts received is indep. NOT: Events indep. If Geo(0.3) stated, can score max B0B0B1B1. If Geo(5, 0.3) stated, can score max B0B1B1B1 |
| 0.8369, 0.8369 – 0.5282 or $^7C_3(0.7)^3(0.3)^2$ = 0.3087 or 0.309 (3 sf) | B1 1, M1, A1 2 | or 0.837 | or B(7, "0.3087") stated or 1 – "0.3087" used instead of "0.3087" |
| $p$ = "0.3087" | M1 | (iib) used in a calc'n eg "0.3087" × 3 or B(7, "0.3087") stated | or 1 – "0.3087" used instead of "0.3087" |
| $^7C_3(1–"0.3087")^4(0.3087)^3$ = 0.235 (3 sf) | M1, A1 3 | | $n$ = 35 or 15: max M1M0A0 |

## Question 6i

| 7! ÷ 3! ÷ 2! or 7! ÷ 2! ÷ 3! = 420 | M1, M1dep, A1 3 | But NOT '${}_7P_4$ or 7!/(7-4)!' if seen | $\frac{7!}{3!+2!}$ M1M0 or $\frac{7!}{3!×n!}$ any $n$: M1M0 |

## Question 6iia

| $^{10}C_5$ or $^{10}C_4$ seen, $^{5}C_x$ × $^{10}C_4$ = 2100 | M1, M1, A1 3 | or 10 or 210 | $^5C_3 × ^{10}C_4$ M1M1A0, anything |

## Question 6b

| $^4C_2 × ^9C_1$ or $^C_3 × ^9C_1$ or 756 or 336 | M1 | $\frac{2}{5}$ or $\frac{4}{10}$ oe. $\frac{2}{5}×(1–\frac{4}{10})$ or $(1–\frac{2}{3})×\frac{4}{10}$. $\frac{2}{5}×(1–\frac{4}{10})+(1–\frac{2}{5})×\frac{4}{10}$ = $\frac{13}{25}$ | $^4C_2 × ^9C_1$ or 60 × 5040 or 302400: SC B1. Not from incorrect wking. SC $\frac{1}{5}×\frac{9}{10}$ or $\frac{4}{5}×\frac{1}{10}$ M1. $\frac{1}{5}×\frac{9}{10}+\frac{4}{5}×\frac{1}{10}$ M1. $(\frac{13}{50}$ A0) |
| $^4C_2 × ^9C_4 + ^C_3 × ^{9}C_3$ or 1092 | M1 | | |
| ÷ 2100 or ÷ (iia) dep ≥ one M1 scored = $\frac{13}{25}$ or 0.52 | M1dep, A1 4 | | Not from incorrect wking. ie P(WA or GA or both) Must be correct figures. ie P(WA or GA but not both) Must be correct figures. SC: $^1P_4 × ^nP_t + ^nP_s × ^nP_t$ ÷ (iia) M1dep. Careful: 336 or 756 can be obtained by incorrect methods. |

## Question 7i

| (0×$a$) + 2×(1 – $a$) = 2 – 2$a$ or 2(1 – $a$) oe | M1, A1 2 | or 2(1 – $a$). Not ISW | Condone 2 × 1 – $a$. Eg E(X) = 2 – 2$a$; 2 – 2$a$ = 1; $a$ = 0.5: M1A0 |

## Question 7ii

| (0×$a$) + 2²×(1 – $a$) – "($2 – 2a$)"² = 4 – 4$a$ – 4 + 8$a$ – 4$a^2$ = 4$a$ – 4$a^2$ (= 4$a$(1 – $a$)) AG | M1, M1, A1 3 | or 4 – 4$a$ oe. – ($i$)² dep contains $a$; ISW; Indep mk or 4(1 – $a$) – 4(1 – $a$)². 4(1 – $a$)(1 – (1 – $a$)). Correct table oe. Var(X) = $a$(-2+2$a$)² + 4$a²$(1 – $a$) M1. $4a^3 – 8a^2 +4a + 4a^2 – 4a^3$, $4a – 4a^2$, A1 | 4 – 4$a$ – 4 + 8$a$ ± 4$a^2$ or 4 – 4$a$ – 4 ± 8$a$ ± 4$a^2$ or equiv M1M1A0. Careful: 4 – 4$a$ – (2 – 2$a$)² = –4 – 4$a$ – (– 4 $a^2$) = –4$a$ + 4$a^2$ = 4$a$(1 – $a$). M1M1A0 only |

## Question 8i

| EDCBA | B1 1 | A 5, B 4, C 3, D 2, E 1 | NOT just 5, 4, 3, 2, 1 |

## Question 8iia

| $1–\frac{65d^2}{5(5^2-1)}$ = 0.9 | M1 | One correct step or better & nothing incorrect for A1. |
| $1–\frac{6o×d^2}{5×24}$ = 0.9 or 0.1 = $\frac{6×d^2}{5×24}$ | A1 2 | ($\Sigma d^2$ = 2 AG) | |

## Question 8b

| $d^2$: 0, 0, 1, any order BACDE or similar | M1, A1 2 | or $d$: 0, 0, 0, 1, -1 any order. Any two adjacent dogs interchanged | May not be seen. If clearly comparing second race with third; DECBA or similar: B1, but must be clear |

## Total marks: 72