O Level Additional Mathematics Statistics Probability Quiz

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O-Level Additional Mathematics Quiz - Statistics Probability

Name: ___________________________
Class: ___________________________
Date: ___________________________
Score: _______ / 40

Duration: 60 minutes
Total Marks: 40

Instructions:

• Answer all 20 questions.
• Show your working clearly where required.
• Section A: 10 short questions (2 marks each). Section B: 6 structured questions (3 marks each). Section C: 4 extended questions (4 marks each).
• Use calculators where appropriate. Give answers correct to 3 significant figures where not exact.

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Section A (Questions 1–10, 2 marks each)

1. A bag contains 5 red and 7 blue marbles. A marble is drawn at random. Find the probability that it is red.

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2. Two fair dice are rolled. Find the probability that the sum of the scores is 7.

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3. Events A and B are such that $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.2$. Find $P(A \cup B)$.

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4. A coin is tossed 3 times. Find the probability of getting exactly 2 heads.

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5. The table shows the number of students in a class by gender and favourite subject.

	Math	Science	Total
Male	8	6	14
Female	10	7	17
Total	18	13	31

A student is chosen at random. Find the probability the student is female given they like Math.

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6. A box has 4 green, 3 yellow, and 5 red pens. Two pens are drawn without replacement. Find the probability both are red.

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7. For a binomial distribution $X \sim B(10, 0.3)$, state the mean of $X$.

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8. Events A and B are independent with $P(A)=0.6$ and $P(B)=0.4$. Find $P(A \cap B)$.

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9. A spinner has 4 equal sectors labelled 1, 2, 3, 4. Find the probability of spinning an even number.

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10. From a deck of 52 cards, one card is drawn. Find the probability it is a heart.

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Section B (Questions 11–16, 3 marks each)

11. A survey of 80 students found 50 like basketball, 30 like badminton, and 10 like both. Find the probability a randomly selected student likes basketball or badminton.

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12. A bag contains 3 white and 2 black balls. Two balls are drawn at random without replacement. Find the probability that exactly one is white.

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13. A random variable $X \sim B(n, p)$ has mean 6 and variance 2.4. Find $n$ and $p$.

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14. The probability that a machine produces a defective item is 0.1. In a batch of 5 items, find the probability exactly 2 are defective. Use $X \sim B(5, 0.1)$.

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15. Events A and B are such that $P(A)=0.5$, $P(B)=0.3$, $P(A|B)=0.6$. Find $P(A \cap B)$ and determine if A and B are independent.

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16. The table shows the probability distribution of a discrete random variable $X$.

$x$	1	2	3	4
$P(X=x)$	0.2	$k$	0.3	0.1

Find $k$ and $E(X)$.

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Section C (Questions 17–20, 4 marks each)

17. A box contains 5 red, 3 blue, and 2 green balls. Three balls are drawn without replacement. Find the probability that they are all of different colours.

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18. In a class of 20 students, 12 take Additional Mathematics and 8 take Physics. 5 take both. A student is chosen at random. Find the probability the student takes Additional Mathematics given they take Physics. Hence find the probability the student takes neither.

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19. A biased coin has $P(\text{head})=0.6$. The coin is tossed 4 times. Let $X$ be the number of heads.
(a) State the distribution of $X$.
(b) Find $P(X \ge 3)$.

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20. The discrete random variable $X$ has probability distribution:

$x$	0	1	2	3
$P(X=x)$	0.1	0.2	$p$	0.3

(a) Find $p$.
(b) Find $E(X)$ and $Var(X)$.

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O-Level Additional Mathematics Quiz - Statistics Probability (Answer Key)

Total Marks: 40
Topic: Statistics Probability

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Section A Answers (Q1–10, 2 marks each)

Q1. Probability = $\frac{5}{5+7} = \frac{5}{12}$.
Teaching note: Total marbles = 12, favourable = 5 red. $P(\text{red}) = \frac{\text{favourable}}{\text{total}}$.
Marks: 1 for total, 1 for final answer.

Q2. Favourable pairs: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 outcomes. Total = 36. $P = \frac{6}{36} = \frac{1}{6}$.
Teaching note: Sum 7 has 6 combinations out of 6×6=36 equally likely outcomes.
Marks: 1 for favourable count, 1 for answer.

Q3. $P(A \cup B) = P(A)+P(B)-P(A\cap B) = 0.4+0.5-0.2 = 0.7$.
Teaching note: Addition rule for union.
Marks: 1 for formula, 1 for answer.

Q4. $X \sim B(3,0.5)$, $P(X=2) = \binom{3}{2}(0.5)^2(0.5)^1 = 3 \times 0.125 = 0.375$.
Teaching note: Exactly 2 heads from 3 tosses.
Marks: 1 for setup, 1 for answer.

Q5. $P(\text{Female}|\text{Math}) = \frac{10}{18} = \frac{5}{9}$.
Teaching note: Conditional: among Math lovers (18), females = 10.
Marks: 1 for denominator, 1 for answer.

Q6. $P(\text{both red}) = \frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$.
Teaching note: Without replacement: second draw has 4 red of 11 left.
Marks: 1 for method, 1 for answer.

Q7. Mean = $np = 10 \times 0.3 = 3$.
Teaching note: Binomial mean formula.
Marks: 2 for correct statement.

Q8. Independent → $P(A\cap B)=P(A)P(B)=0.6\times0.4=0.24$.
Teaching note: Independence definition.
Marks: 2 for answer.

Q9. Even sectors: 2,4 → 2 of 4. $P = \frac{2}{4}=0.5$.
Marks: 2.

Q10. Hearts = 13 of 52. $P = \frac{13}{52}=\frac{1}{4}$.
Marks: 2.

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Section B Answers (Q11–16, 3 marks each)

Q11. $P(B \cup M) = \frac{50+30-10}{80} = \frac{70}{80} = \frac{7}{8}$.
Teaching note: Inclusion-exclusion on 80 students.
Marks: 1 count, 1 subtraction, 1 answer.

Q12. Exactly one white: (W,B) or (B,W).
$P = \frac{3}{5}\times\frac{2}{4} + \frac{2}{5}\times\frac{3}{4} = \frac{6}{20}+\frac{6}{20}=\frac{12}{20}=0.6$.
Marks: 1 for cases, 1 for calc, 1 answer.

Q13. Mean $np=6$, Var $np(1-p)=2.4$. So $6(1-p)=2.4 \Rightarrow 1-p=0.4 \Rightarrow p=0.6$. Then $n=10$.
Marks: 1 eqn, 1 p, 1 n.

Q14. $P(X=2)=\binom{5}{2}(0.1)^2(0.9)^3 = 10 \times 0.01 \times 0.729 = 0.0729$.
Marks: 1 formula, 1 sub, 1 answer.

Q15. $P(A\cap B)=P(A|B)P(B)=0.6\times0.3=0.18$. Independent if $P(A\cap B)=P(A)P(B)=0.15$; since 0.18≠0.15, NOT independent.
Marks: 1 calc, 1 compare, 1 conclusion.

Q16. $\sum P =1 \Rightarrow 0.2+k+0.3+0.1=1 \Rightarrow k=0.4$.
$E(X)=1(0.2)+2(0.4)+3(0.3)+4(0.1)=0.2+0.8+0.9+0.4=2.3$.
Marks: 1 k, 1 E setup, 1 answer.

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Section C Answers (Q17–20, 4 marks each)

Q17. Total ways = $\binom{10}{3}=120$. Favourable: 1 red,1 blue,1 green = $5\times3\times2=30$. $P=\frac{30}{120}=0.25$.
Marks: 1 total, 1 favourable, 1 division, 1 answer.

Q18. $P(AM|P)=\frac{5}{8}=0.625$. Neither = 20-(12+8-5)=5, so $P=\frac{5}{20}=0.25$.
Marks: 1 cond, 1 neither count, 1 each prob.

Q19. (a) $X \sim B(4,0.6)$. (b) $P(X\ge3)=P(X=3)+P(X=4)=\binom{4}{3}(0.6)^3(0.4)+\binom{4}{4}(0.6)^4 = 4\times0.216\times0.4 + 0.1296 = 0.3456+0.1296=0.4752$.
Marks: 1 dist, 1 setup, 1 calc, 1 answer.

Q20. (a) $0.1+0.2+p+0.3=1 \Rightarrow p=0.4$. (b) $E=0(0.1)+1(0.2)+2(0.4)+3(0.3)=0+0.2+0.8+0.9=1.9$. $E(X^2)=0+0.2+4(0.4)+9(0.3)=0.2+1.6+2.7=4.5$. $Var=4.5-(1.9)^2=4.5-3.61=0.89$.
Marks: 1 p, 1 E, 1 E(X²), 1 Var.