O Level Additional Mathematics Statistics Probability Quiz

O-Level Additional Mathematics Quiz - Statistics Probability

Answer Key and Marking Scheme

Total Marks: 50

Section A: Basic Probability Concepts (Questions 1–5)

1. P(not blue) = 1 − P(blue) = 1 − 3/10 = 7/10 or 0.7
Answer: 7/10 or 0.7 [2]
Marking: M1 for identifying total = 10 and blue = 3; A1 for correct answer.

2. A = {2, 4, 6}, B = {5, 6}, A ∩ B = {6}
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 3/6 + 2/6 − 1/6 = 4/6 = 2/3
Answer: 2/3 [2]
Marking: M1 for correct formula; A1 for correct answer.

3. For independence, P(X ∩ Y) = P(X) × P(Y).
P(X) × P(Y) = 0.4 × 0.5 = 0.2
Since P(X ∩ Y) = 0.2 = P(X) × P(Y), X and Y are independent.
Answer: Yes, they are independent because P(X ∩ Y) = P(X) × P(Y) = 0.2. [2]
Marking: M1 for calculating P(X) × P(Y); A1 for correct conclusion with justification.

4. P(King or Heart) = P(King) + P(Heart) − P(King of Hearts)
= 4/52 + 13/52 − 1/52 = 16/52 = 4/13
Answer: 4/13 [2]
Marking: M1 for addition rule; A1 for correct answer.

5. P(passes at least one) = P(A ∪ P) = P(A) + P(P) − P(A ∩ P)
= 0.7 + 0.6 − 0.45 = 0.85
Answer: 0.85 [2]
Marking: M1 for correct formula; A1 for correct answer.

Section B: Permutations and Combinations (Questions 6–10)

6. Number of 4-digit numbers = ⁶P₄ = 6 × 5 × 4 × 3 = 360
Answer: 360 [3]
Marking: M1 for recognising permutation; M1 for correct calculation; A1 for correct answer.

7. Choose 2 women from 5: ⁵C₂ = 10
Choose 2 men from 7: ⁷C₂ = 21
Total ways = 10 × 21 = 210
Answer: 210 [3]
Marking: M1 for ⁵C₂; M1 for ⁷C₂ and multiplication; A1 for correct answer.

8. "STATISTICS" has 10 letters: S (3), T (3), A (1), I (2), C (1)
Number of arrangements = 10! / (3! × 3! × 2!) = 3,628,800 / (6 × 6 × 2) = 3,628,800 / 72 = 50,400
Answer: 50,400 [3]
Marking: M1 for identifying repeated letters; M1 for correct formula; A1 for correct answer.

9. Choose 1 goalkeeper from 3: ³C₁ = 3
Choose 4 outfield players from 7: ⁷C₄ = 35
Total teams = 3 × 35 = 105
Answer: 105 [3]
Marking: M1 for ³C₁; M1 for ⁷C₄ and multiplication; A1 for correct answer.

10. Total arrangements without restriction = 6! = 720
Arrangements where A and B sit together: treat AB as one block → 5! × 2! = 120 × 2 = 240
Arrangements where A and B are NOT together = 720 − 240 = 480
Answer: 480 [3]
Marking: M1 for total arrangements; M1 for arrangements with A and B together; A1 for correct answer.

Section C: Probability Distributions and Expectation (Questions 11–15)

11. Sum of probabilities = 1
0.2 + 0.3 + k + 0.1 = 1 → 0.6 + k = 1 → k = 0.4
Answer: k = 0.4 [3]
Marking: M1 for setting sum = 1; M1 for correct equation; A1 for correct answer.

12. E(X) = Σ x·P(X = x) = 1(0.2) + 2(0.3) + 3(0.4) + 4(0.1)
= 0.2 + 0.6 + 1.2 + 0.4 = 2.4
Answer: 2.4 [3]
Marking: M1 for correct formula; M1 for correct substitution; A1 for correct answer.

13. X ~ B(4, 0.5)
P(X = 2) = ⁴C₂ (0.5)² (0.5)² = 6 × 0.25 × 0.25 = 6 × 0.0625 = 0.375
Answer: 0.375 or 3/8 [3]
Marking: M1 for binomial formula; M1 for correct substitution; A1 for correct answer.

14. X ~ B(5, 0.25)
P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5)
P(X = 3) = ⁵C₃ (0.25)³ (0.75)² = 10 × 0.015625 × 0.5625 = 0.08789
P(X = 4) = ⁵C₄ (0.25)⁴ (0.75)¹ = 5 × 0.00390625 × 0.75 = 0.01465
P(X = 5) = ⁵C₅ (0.25)⁵ (0.75)⁰ = 1 × 0.0009766 × 1 = 0.0009766
P(X ≥ 3) ≈ 0.08789 + 0.01465 + 0.0009766 = 0.1035 ≈ 0.104
Answer: 0.104 (3 s.f.) [3]
Marking: M1 for identifying binomial; M1 for correct calculation of at least one term; A1 for correct answer to 3 s.f.

15. Var(X) = np(1 − p) = 8 × 0.3 × 0.7 = 8 × 0.21 = 1.68
Answer: 1.68 [3]
Marking: M1 for correct formula; M1 for correct substitution; A1 for correct answer.

Section D: Data Analysis and Applied Probability (Questions 16–20)

16. Σfx = 0(4) + 1(7) + 2(5) + 3(3) + 4(1) = 0 + 7 + 10 + 9 + 4 = 30
Σf = 20
Mean = 30/20 = 1.5
Answer: 1.5 [3]
Marking: M1 for Σfx; M1 for division by Σf; A1 for correct answer.

17. Σfx² = 0²(4) + 1²(7) + 2²(5) + 3²(3) + 4²(1) = 0 + 7 + 20 + 27 + 16 = 70
Variance = Σfx²/Σf − (mean)² = 70/20 − (1.5)² = 3.5 − 2.25 = 1.25
Standard deviation = √1.25 ≈ 1.118 ≈ 1.12 (3 s.f.)
Answer: 1.12 (3 s.f.) [3]
Marking: M1 for Σfx²; M1 for variance formula; A1 for correct answer to 3 s.f.

18. P(both defective) = (3/8) × (2/7) = 6/56 = 3/28 ≈ 0.107
Answer: 3/28 or 0.107 (3 s.f.) [3]
Marking: M1 for first probability 3/8; M1 for second probability 2/7 and multiplication; A1 for correct answer.

19. X ~ B(10, 0.85)
P(X = 8) = ¹⁰C₈ (0.85)⁸ (0.15)²
¹⁰C₈ = ¹⁰C₂ = 45
(0.85)⁸ ≈ 0.2725, (0.15)² = 0.0225
P(X = 8) = 45 × 0.2725 × 0.0225 ≈ 0.2759 ≈ 0.276
Answer: 0.276 (3 s.f.) [3]
Marking: M1 for binomial formula; M1 for correct substitution; A1 for correct answer to 3 s.f.

20. Tree diagram:

• First draw: P(White) = 4/10 = 2/5, P(Black) = 6/10 = 3/5
• Second draw:
  • If first White: P(White) = 3/9 = 1/3, P(Black) = 6/9 = 2/3
  • If first Black: P(White) = 4/9, P(Black) = 5/9

P(different colours) = P(W then B) + P(B then W)
= (2/5 × 2/3) + (3/5 × 4/9) = 4/15 + 12/45 = 4/15 + 4/15 = 8/15
Answer: 8/15 [3]
Marking: M1 for correct tree diagram probabilities; M1 for identifying both paths and adding; A1 for correct answer.

END OF ANSWER KEY