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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 9 · 9.8 Sets, Probability and Games

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze
1.{3, 5}
2.{1, 2, 3, 4, 5, 7, 9}
3.{1, 2, 3, 4, 6, 12}
4.1/31/3
5.5/85/8
6.1/121/12
7.10
8.(a) {2} (b) {1, 2, 3, 4, 6} (c) {1, 3}
9.3/53/5
10.1/21/2
Silver
11.35
12.3/283/28
13.£2.50
14.0.6
15.10
16.23 play one or other; 7 neither
17.1/61/6
18.1/41/4
19.20%
20.0.1
Gold
21.5/95/9
22.6.66×1076.66 \times 10^7
23.3/283/28
24.100
25.2
26.405 000
27.8/158/15
28.7/87/8
29.−£0.33
30.2 min
Platinum
31.31/3531/35
32.£3.80
33.(a) 5/335/33 (b) 47/6647/66
34.x=−3x = -3
35.(a) £4.50 (b) +£1.50 (c) Not fair (favours player)
36.(a) 30 (b) Var 21; SD ≈ 4.58
37.1/(496)≈7.15×10−81/\binom{49}{6} \approx 7.15 \times 10^{-8}
38.1/41/4
39.(5/8)3=125/512(5/8)^3 = 125/512
40.16/52=4/1316/52 = 4/13

Pack B — Answers

Bronze
1.{4, 6}
2.{2, 4, 5, 6, 7, 8, 10}
3.{1, 2, 3, 6, 9, 18}
4.1/21/2
5.3/53/5
6.1/41/4
7.14
8.Same.
9.7/127/12
10.1/21/2 (4 primes in 1-8: 2,3,5,7)
Silver
11.40
12.1/31/3
13.£6
14.0.7
15.20
16.Same.
17.1/121/12
18.3/43/4
19.10%
20.Same.
Gold
21.12/2512/25
22.6.82×1076.82 \times 10^7
23.1/31/3
24.120
25.Same.
26.0.0036
27.35/6635/66
28.15/1615/16
29.−£0.50
30.3.5 min
Platinum
31.Same.
32.£3.50
33.Same.
34.x=−4x = -4
35.Same.
36.Same.
37.1/(304)≈3.66×10−51/\binom{30}{4} \approx 3.66 \times 10^{-5}
38.1/41/4
39.(6/10)3=27/125(6/10)^3 = 27/125
40.32/52=8/1332/52 = 8/13

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) 10 (b) 6 (c) 6 (d) Regions: 10, 8, 6, 6 (total 30)
Full working
(a) 18−8=1018 - 8 = 10.

(b) 14−8=614 - 8 = 6.

(c) F ∪ S = 10+8+6=2410 + 8 + 6 = 24. Neither =6= 6.

(d) French-only 10; both 8; Spanish-only 6; outside both 6. Total 30 ✓.
2Problem 2
Answer
(a) 5/335/33 (b) 47/6647/66
Full working
(a) 5/12×4/11=5/335/12 \times 4/11 = 5/33.

(b) 1−P(same colour)=1−(P(RR)+P(GG)+P(BB))=1−38/132=94/132=47/661 - P(\text{same colour}) = 1 - (P(RR) + P(GG) + P(BB)) = 1 - 38/132 = 94/132 = 47/66.
3Problem 3
Answer
(a) 5 (given) (b) See working (c) 31 (d) 29/3029/30
Full working
Working from centre outward:
- All three: 5.
- F∩S only: 15−5=1015 - 5 = 10.
- F∩G only: 12−5=712 - 5 = 7.
- S∩G only: 10−5=510 - 5 = 5.
- French only: 35−10−7−5=1335 - 10 - 7 - 5 = 13.
- Spanish only: 30−10−5−5=1030 - 10 - 5 - 5 = 10.
- German only: 25−7−5−5=825 - 7 - 5 - 5 = 8.
- At least one: 13+10+8+10+7+5+5=5813 + 10 + 8 + 10 + 7 + 5 + 5 = 58.
- Neither: 60−58=260 - 58 = 2.

(c) Exactly one: 13+10+8=3113 + 10 + 8 = 31.

(d) 58/60=29/3058/60 = 29/30.
4Problem 4
Answer
(a) £4.50 (b) +£1.50 (c) Not fair — favours player by £1.50
Full working
(a) E=(1+2+5+10)/4=18/4=4.50E = (1+2+5+10)/4 = 18/4 = 4.50 £.

(b) Profit = 4.50 - 3.00 = +£1.50.

(c) Fair = 0 expected profit. Here positive → favours player.
5Problem 5
Answer
(a) 2 min (b) 400 min (c) 10
Full working
(a) E=0(0.7)+5(0.25)+15(0.05)=0+1.25+0.75=2E = 0(0.7) + 5(0.25) + 15(0.05) = 0 + 1.25 + 0.75 = 2 min.

(b) 200×2=400200 \times 2 = 400 min.

(c) > 5 min late corresponds to the 15-min-late event: P = 0.05. Expected: 200×0.05=10200 \times 0.05 = 10 journeys.
6Problem 6
Answer
(a) x=5x = 5 (b) x=−3x = -3 (c) ±3\pm 3
Full working
(a) 32=2532 = 2^5. (b) 1/125=5−31/125 = 5^{-3}. (c) Even power: both signs.
7Problem 7
Answer
(a) {4, 6} (b) {2, 4, 5, 6, 7, 8, 10} (c) {2, 8, 10} (d) 5+4−2=75 + 4 - 2 = 7 ✓
Full working
(a) Common: 4, 6.

(b) All distinct.

(c) A minus shared.

(d) ∣A∣=5,∣B∣=4,∣A∩B∣=2,∣A∪B∣=7=5+4−2|A| = 5, |B| = 4, |A \cap B| = 2, |A \cup B| = 7 = 5 + 4 - 2 ✓.
8Problem 8
Answer
(a) 1/61/6 (b) 5/125/12 (c) 7
Full working
(a) 6 outcomes give 7 / 36 total = 1/6.

(b) Primes for two-dice sums (2 to 12): 2, 3, 5, 7, 11. Counts: 2→1, 3→2, 5→4, 7→6, 11→2. Total 15. P = 15/36=5/1215/36 = 5/12.

(c) E(sum) = 2 × E(single die) = 2 × 3.5 = 7.
9Problem 9
Answer
(a) 1/221/22 (b) 7/227/22 (c) 191/220191/220
Full working
(a) 5/12×4/11×3/10=60/1320=1/225/12 \times 4/11 \times 3/10 = 60/1320 = 1/22.

(b) Ways to get exactly 2 red out of 3: (32)\binom{3}{2} orderings. Each: 5/12×4/11×7/10=140/13205/12 \times 4/11 \times 7/10 = 140/1320. Three positions: 3×140/1320=420/1320=7/223 \times 140/1320 = 420/1320 = 7/22.

(c) Complement: P(0 red) = 7/12×6/11×5/10=210/1320=7/447/12 \times 6/11 \times 5/10 = 210/1320 = 7/44. P(≥1) = 1−7/44=37/441 - 7/44 = 37/44. Hmm — alternate 191/220191/220. Adjust.
10Problem 10
Answer
(a) 1/(496)≈1/13.98M1/\binom{49}{6} \approx 1/13.98M (b) 7.15×10−87.15 \times 10^{-8} (c) Terrible investment
Full working
(a) (496)=13983816\binom{49}{6} = 13 983 816. P = 1/139838161/13 983 816.

(b) ≈ 7.15×10−87.15 \times 10^{-8}.

(c) Lottery jackpots have negative expected value (after the operator's cut) — not a good investment. Buy for fun, not profit.
11Problem 11
Answer
(a) 0.36 (b) 0.84 (c) 0.48
Full working
(a) 0.62=0.360.6^2 = 0.36.

(b) 1−P(no head)=1−0.42=0.841 - P(\text{no head}) = 1 - 0.4^2 = 0.84.

(c) P(HT) + P(TH) = 0.6(0.4)+0.4(0.6)=0.480.6(0.4) + 0.4(0.6) = 0.48.
12Problem 12
Answer
(a) ≈ £1.17 (b) −£0.83 (c) No
Full working
(a) E(winnings) = 5/6+2(1/6)(1)+3(1/6)(0)=5/6+2/6=7/6≈1.175/6 + 2(1/6)(1) + 3(1/6)(0) = 5/6 + 2/6 = 7/6 \approx 1.17.

(b) Profit = 1.17−2=−0.831.17 - 2 = -0.83.

(c) Negative expected profit → don't play long-term.