A Level H2 Mathematics Statistics Probability Quiz

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A-Level Maths H2 Quiz - Statistics Probability

Name: ________________________
Class: ________________________
Date: ________________________
Score: _________ / 50

Duration: 60 minutes
Total Marks: 50

Instructions:

• Answer all 20 questions.
• Show your working clearly where required.
• Use a graphing calculator where appropriate.
• Write your answers in the spaces provided.

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Section A: Probability (Questions 1–7) [17 marks]

1. [2 marks]
A bag contains 5 red balls and 3 green balls. Two balls are drawn at random without replacement. Find the probability that both balls are red.

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2. [2 marks]
Events A and B are such that $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.2$. Determine whether A and B are independent. Give a reason.

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3. [3 marks]
A fair die is rolled twice. Let X be the sum of the two scores. Find $P(X = 7)$.

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4. [2 marks]
In a class of 30 students, 18 study Chemistry and 12 study Physics. If 5 study both, find the probability that a randomly chosen student studies Chemistry or Physics.

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5. [3 marks]
A box contains 4 blue, 3 yellow, and 5 white marbles. Three marbles are drawn at random without replacement. Find the probability that exactly two are white.

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6. [2 marks]
The probability that a machine produces a defective item is 0.05. Items are inspected independently. Find the probability that in a sample of 10 items, exactly 2 are defective.

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7. [3 marks]
Two events C and D are mutually exclusive with $P(C) = 0.3$ and $P(D) = 0.4$. Find $P(C \cup D)$ and $P(C' \cap D')$.

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Section B: Discrete Random Variables (Questions 8–12) [13 marks]

8. [3 marks]
A discrete random variable X has probability distribution:

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

Find the value of $k$ and $E(X)$.

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9. [2 marks]
For the random variable X in Q8, find $Var(X)$.

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10. [3 marks]
A biased coin has probability 0.6 of landing heads. Let Y be the number of heads in 4 tosses. Find $P(Y \ge 3)$ and $E(Y)$.

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11. [2 marks]
The random variable W has $E(W) = 5$ and $Var(W) = 4$. Find $E(2W + 3)$ and $Var(2W + 3)$.

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12. [3 marks]
A game costs $2 to play. A player wins$5 if a fair die shows 6, and wins nothing otherwise. Let P be the profit. Find the probability distribution of P and $E(P)$.

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Section C: Data Analysis & Regression (Questions 13–20) [20 marks]

13. [2 marks]
The masses (in kg) of 8 students are: 52, 55, 48, 60, 57, 53, 49, 58. Find the mean and standard deviation.

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14. [2 marks]
For the data in Q13, find the median and interquartile range.

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15. [3 marks]
The following table shows the number of hours studied ($x$) and test score ($y$) for 6 students:

$x$	2	3	5	6	8	9
$y$	50	55	65	70	78	85

Find the product moment correlation coefficient $r$, correct to 3 decimal places.

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16. [3 marks]
Using the data in Q15, find the equation of the regression line of $y$ on $x$ in the form $y = a + bx$, correct to 2 decimal places.

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17. [2 marks]
Using the regression line from Q16, estimate the test score for a student who studies 7 hours.

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18. [2 marks]
State what is meant by a random sample in the context of surveying students in a school.

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19. [3 marks]
The times (in minutes) taken by 10 students to complete a task are: 12, 15, 14, 18, 20, 13, 16, 17, 19, 11. Draw a box plot.

Image pending generation: diagram for Q19.

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20. [3 marks]
A survey of 200 residents records whether they exercise regularly and their age group. The results are:

	Exercise	No Exercise	Total
Under 30	60	40	100
30 and over	50	50	100
Total	110	90	200

Find the probability that a randomly selected resident exercises regularly given they are under 30. Hence, comment on whether age group and exercise are independent.

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A-Level Maths H2 Quiz - Statistics Probability: Answer Key

Total Marks: 50
Topic: Statistics & Probability

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Section A: Probability

Q1 [2 marks]
Total balls = 8. $P(\text{first red}) = \frac{5}{8}$. After one red removed, 4 red, 7 total: $P(\text{second red}) = \frac{4}{7}$.
$P(\text{both red}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$.
Teaching note: Without replacement means the total and favourable counts decrease.
Common mistake: Using $\frac{5}{8} \times \frac{5}{8}$.

Q2 [2 marks]
Independent if $P(A \cap B) = P(A)P(B)$.
$P(A)P(B) = 0.4 \times 0.5 = 0.2 = P(A \cap B)$.
Thus A and B are independent.
Teaching note: Definition of independence for events.

Q3 [3 marks]
Total outcomes = $6 \times 6 = 36$. Sum 7 occurs as (1,6),(2,5),(3,4),(4,3),(5,2),(6,1): 6 ways.
$P(X=7) = \frac{6}{36} = \frac{1}{6}$.
Mark breakdown: 1 mark total outcomes, 2 marks for correct count and probability.

Q4 [2 marks]
$P(C \cup P) = P(C) + P(P) - P(C \cap P) = \frac{18}{30} + \frac{12}{30} - \frac{5}{30} = \frac{25}{30} = \frac{5}{6}$.

Q5 [3 marks]
Total marbles = 12. Choose 3: $\binom{12}{3} = 220$. Exactly 2 white: $\binom{5}{2}\binom{7}{1} = 10 \times 7 = 70$.
$P = \frac{70}{220} = \frac{7}{22}$.
Marking: 1 mark combinations, 2 marks final answer.

Q6 [2 marks]
$X \sim B(10, 0.05)$. $P(X=2) = \binom{10}{2}(0.05)^2(0.95)^8 = 45 \times 0.0025 \times 0.6634 \approx 0.0746$.

Q7 [3 marks]
Mutually exclusive $\Rightarrow P(C \cup D) = P(C)+P(D) = 0.3+0.4 = 0.7$.
$P(C' \cap D') = P((C \cup D)') = 1 - 0.7 = 0.3$.
Marking: 1 mark each part.

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Section B: Discrete Random Variables

Q8 [3 marks]
$\sum P = 1 \Rightarrow 0.1 + k + 0.3 + 0.2 = 1 \Rightarrow k = 0.4$.
$E(X) = 1(0.1)+2(0.4)+3(0.3)+4(0.2) = 0.1+0.8+0.9+0.8 = 2.6$.
Marking: 1 mark k, 2 marks E(X).

Q9 [2 marks]
$E(X^2) = 1(0.1)+4(0.4)+9(0.3)+16(0.2) = 0.1+1.6+2.7+3.2 = 7.6$.
$Var(X) = 7.6 - (2.6)^2 = 7.6 - 6.76 = 0.84$.

Q10 [3 marks]
$Y \sim B(4, 0.6)$. $P(Y\ge3) = P(Y=3)+P(Y=4) = \binom{4}{3}(0.6)^3(0.4) + (0.6)^4 = 4(0.216)(0.4)+0.1296 = 0.3456+0.1296 = 0.4752$.
$E(Y) = 4 \times 0.6 = 2.4$.
Marking: 2 marks probability, 1 mark expectation.

Q11 [2 marks]
$E(2W+3) = 2E(W)+3 = 10+3 = 13$.
$Var(2W+3) = 4Var(W) = 16$.

Q12 [3 marks]
Profit P: if 6 shows (prob 1/6), P = 5-2 = 3; else P = -2 (prob 5/6).
Distribution: $P(P=3)=1/6$, $P(P=-2)=5/6$.
E(P) = 3(1/6) + (-2)(5/6) = 0.5 - 1.6667 = -1.1667 \approx -\1.17$.
Marking: 1 mark table, 2 marks E(P).

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Section C: Data Analysis & Regression

Q13 [2 marks]
Mean = $(52+55+48+60+57+53+49+58)/8 = 432/8 = 54$ kg.
Variance = $\frac{\sum x^2}{n} - \bar{x}^2 = \frac{23548}{8} - 2916 = 2943.5 - 2916 = 27.5$. SD = $\sqrt{27.5} \approx 5.24$ kg.

Q14 [2 marks]
Ordered: 48,49,52,53,55,57,58,60. Median = (53+55)/2 = 54.
Q1 = (49+52)/2 = 50.5, Q3 = (57+58)/2 = 57.5. IQR = 7.

Q15 [3 marks]
$\bar{x} = 33/6 = 5.5$, $\bar{y} = 403/6 \approx 67.167$.
$S_{xx} = 199 - 6(5.5)^2 = 199 - 181.5 = 17.5$.
$S_{yy} = 27763 - 6(67.167)^2 \approx 1044.83$.
$S_{xy} = 2419 - 6(5.5)(67.167) = 2419 - 2214.5 = 204.5$.
$r = \frac{204.5}{\sqrt{17.5 \times 1044.83}} \approx 0.989$.
Marking: 1 mark means, 2 marks r.

Q16 [3 marks]
$b = S_{xy}/S_{xx} = 204.5/17.5 \approx 11.69$.
$a = \bar{y} - b\bar{x} = 67.167 - 11.69(5.5) \approx 2.85$.
$y = 2.85 + 11.69x$.

Q17 [2 marks]
$x=7 \Rightarrow y = 2.85 + 11.69(7) = 84.68 \approx 85$.

Q18 [2 marks]
A random sample means every student in the school has an equal chance of being selected, and selections are independent.
Teaching note: Avoid bias from convenience sampling.

Q19 [2 marks]
Ordered: 11,12,13,14,15,16,17,18,19,20.
Min=11, Q1=(13+14)/2=13.5, Median=(15+16)/2=15.5, Q3=(18+19)/2=18.5 (using linear interpolation method; accepted 18), Max=20.
Box plot as per placeholder: axis 10–21, box 13.5 to 18.5, median line at 15.5, whiskers to 11 and 20.

Q20 [3 marks]
$P(\text{Exercise} \mid \text{Under 30}) = 60/100 = 0.6$.
$P(\text{Exercise}) = 110/200 = 0.55$. Since $0.6 \neq 0.55$, not independent.
Marking: 1 mark conditional, 2 marks comment.