Secondary 4 Additional Mathematics Statistics Probability Quiz

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Secondary 4 Additional Mathematics Quiz - Statistics Probability

Name: __________________________
Class: __________________________
Date: __________________________
Score: ________ / 65

Duration: 90 Minutes
Total Marks: 65

Instructions:

• Answer all questions.
• Show all necessary working clearly.
• Use a scientific calculator where appropriate.
• Give your answers to 3 significant figures unless stated otherwise.

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Section A: Probability (Questions 1–10)

1. A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. Find the probability that both balls are of the same colour.
   
   
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2. The probability that student A passes a test is 0.7 and the probability that student B passes is 0.6. Given that the events are independent, find the probability that at least one of them passes.
   
   
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3. In a group of 100 students, 60 like Mathematics, 50 like Physics, and 30 like both. If a student is chosen at random, find the probability that they like neither Mathematics nor Physics.
   
   
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4. A fair six-sided die is rolled twice. Let $X$ be the sum of the two scores. Find $P(X > 9)$.
   
   
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5. Given $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cup B) = 0.7$. Find $P(A \cap B)$.
   
   
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6. A box contains 8 light bulbs, of which 3 are defective. Three bulbs are selected at random without replacement. Find the probability that exactly one bulb is defective.
   
   
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7. Two events $E$ and $F$ are such that $P(E) = 0.3$ and $P(F|E) = 0.4$. Find $P(E \cap F)$.
   
   
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8. A target is shot at by two archers. The probability that Archer 1 hits the target is $\frac{2}{3}$ and for Archer 2 is $\frac{3}{4}$. Find the probability that only one of them hits the target.
   
   
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9. A card is drawn from a standard deck of 52 cards. Find the probability that the card is either a Heart or a King.
   
   
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10. In a probability distribution, the possible values of $X$ are 1, 2, 3, and 4. The probabilities are $P(X=1)=0.1$, $P(X=2)=0.3$, $P(X=3)=k$, and $P(X=4)=0.2$. Find the value of $k$ and calculate $E(X)$.
    
    
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Section B: Statistics (Questions 11–20)

11. The heights of a group of plants are normally distributed with a mean of 15 cm and a standard deviation of 2 cm. Find the probability that a randomly chosen plant is taller than 18 cm.
    
    
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12. For the normal distribution in Question 11, find the range of heights that contains the middle 95% of the plants.
    
    
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13. A set of data has a mean of 50 and a variance of 25. If every value in the set is multiplied by 2 and then increased by 5, find the new mean and new variance.
    
    
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14. The weights of 10 samples are: 45, 48, 50, 52, 55, 55, 58, 60, 62, 75. Calculate the mean and the standard deviation.
    
    
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15. A random variable $X$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$. Given that $P(X < 10) = 0.2$ and $P(X < 20) = 0.8$, set up the two simultaneous equations to find $\mu$ and $\sigma$. (Do not solve).
    
    
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16. In a normal distribution, 10% of the data lies above 85 and 10% lies below 45. Find the mean and standard deviation of this distribution.
    
    
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17. A discrete random variable $Y$ has the following probability distribution: $Y=0, P(Y=0)=0.2$ $Y=1, P(Y=1)=0.5$ $Y=2, P(Y=2)=0.3$ Calculate the variance $Var(Y)$.
    
    
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18. The scores of a class in a test are normally distributed. The mean is 62 and the standard deviation is 8. If the passing mark is 50, what percentage of the class failed the test?
    
    
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19. A continuous random variable $X$ has a probability density function $f(x)$ defined on $[0, 2]$. If $f(x) = kx$ for $0 \le x \le 2$ and $f(x) = 0$ otherwise, find the value of $k$.
    
    
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20. For the density function in Question 19, calculate the expected value $E(X)$.
    
    
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Answer Key - Statistics Probability Quiz

1. Probability of same colour: $P(RR) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56}$ $P(BB) = \frac{3}{8} \times \frac{2}{7} = \frac{6}{56}$ Total $= \frac{26}{56} = \frac{13}{28} \approx 0.464$ [3 marks]

2. At least one passes: $P(\text{at least one}) = 1 - P(\text{both fail})$ $P(\text{both fail}) = (1 - 0.7) \times (1 - 0.6) = 0.3 \times 0.4 = 0.12$ $P = 1 - 0.12 = 0.88$ [3 marks]

3. Neither Math nor Physics: $P(M \cup P) = P(M) + P(P) - P(M \cap P) = 0.6 + 0.5 - 0.3 = 0.8$ $P(\text{neither}) = 1 - 0.8 = 0.2$ [3 marks]

4. Sum $X > 9$: Possible outcomes: (4,6), (5,5), (6,4), (5,6), (6,5), (6,6) $\rightarrow 6$ outcomes. Total outcomes $= 36$. $P = \frac{6}{36} = \frac{1}{6} \approx 0.167$ [3 marks]

5. $P(A \cap B)$: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ $0.7 = 0.4 + 0.5 - P(A \cap B) \implies P(A \cap B) = 0.2$ [2 marks]

6. Exactly one defective: Ways to choose 1 defective and 2 good: $\binom{3}{1} \times \binom{5}{2} = 3 \times 10 = 30$ Total ways: $\binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$ $P = \frac{30}{56} = \frac{15}{28} \approx 0.536$ [4 marks]

7. $P(E \cap F)$: $P(F|E) = \frac{P(E \cap F)}{P(E)} \implies P(E \cap F) = 0.4 \times 0.3 = 0.12$ [2 marks]

8. Only one hits: $P(\text{1 hits, 2 misses}) + P(\text{1 misses, 2 hits})$ $= (\frac{2}{3} \times \frac{1}{4}) + (\frac{1}{3} \times \frac{3}{4}) = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \approx 0.417$ [3 marks]

9. Heart or King: $P(H) = \frac{13}{52}$, $P(K) = \frac{4}{52}$, $P(H \cap K) = \frac{1}{52}$ $P = \frac{13+4-1}{52} = \frac{16}{52} = \frac{4}{13} \approx 0.308$ [3 marks]

10. $k$ and $E(X)$: $\sum P = 1 \implies 0.1 + 0.3 + k + 0.2 = 1 \implies k = 0.4$ $E(X) = (1 \times 0.1) + (2 \times 0.3) + (3 \times 0.4) + (4 \times 0.2) = 0.1 + 0.6 + 1.2 + 0.8 = 2.7$ [4 marks]

11. $P(X > 18)$: $z = \frac{18 - 15}{2} = 1.5$ $P(Z > 1.5) = 1 - P(Z < 1.5) = 1 - 0.9332 = 0.0668$ [3 marks]

12. Middle 95%: $z = \pm 1.96$ $x = \mu \pm z\sigma = 15 \pm 1.96(2) = 15 \pm 3.92$ Range: $11.08 \text{ cm to } 18.92 \text{ cm}$ [4 marks]

13. New Mean and Variance: New Mean $= (50 \times 2) + 5 = 105$ New Variance $= 2^2 \times 25 = 4 \times 25 = 100$ [4 marks]

14. Mean and SD: Mean $= \frac{45+48+50+52+55+55+58+60+62+75}{10} = \frac{560}{10} = 56$ $\text{Variance} = \frac{\sum (x-\bar{x})^2}{n} = \frac{11^2+8^2+6^2+4^2+1^2+1^2+2^2+4^2+6^2+19^2}{10} = \frac{121+64+36+16+1+1+4+16+36+361}{10} = \frac{656}{10} = 65.6$ $\text{SD} = \sqrt{65.6} \approx 8.10$ [4 marks]

15. Simultaneous Equations: $z_1 = \frac{10 - \mu}{\sigma} = -0.842$ (from $P=0.2$) $z_2 = \frac{20 - \mu}{\sigma} = 0.842$ (from $P=0.8$) Equations: $10 - \mu = -0.842\sigma$ and $20 - \mu = 0.842\sigma$ [4 marks]

16. Mean and SD: $\mu$ is midpoint of 45 and 85 $\implies \mu = \frac{45+85}{2} = 65$ $z = \frac{85 - 65}{\sigma} = 1.282$ (for top 10%) $\sigma = \frac{20}{1.282} \approx 15.6$ [5 marks]

17. $Var(Y)$: $E(Y) = (0 \times 0.2) + (1 \times 0.5) + (2 \times 0.3) = 0 + 0.5 + 0.6 = 1.1$ $E(Y^2) = (0^2 \times 0.2) + (1^2 \times 0.5) + (2^2 \times 0.3) = 0 + 0.5 + 1.2 = 1.7$ $Var(Y) = 1.7 - (1.1)^2 = 1.7 - 1.21 = 0.49$ [4 marks]

18. Percentage failed: $z = \frac{50 - 62}{8} = -1.5$ $P(Z < -1.5) = 0.0668$ Percentage $= 6.68\%$ [4 marks]

19. Value of $k$: $\int_0^2 kx \, dx = 1 \implies [ \frac{1}{2}kx^2 ]_0^2 = 1 \implies \frac{1}{2}k(4) = 1 \implies 2k = 1 \implies k = 0.5$ [4 marks]

20. $E(X)$: $E(X) = \int_0^2 x(0.5x) \, dx = \int_0^2 0.5x^2 \, dx = [ \frac{0.5x^3}{3} ]_0^2 = \frac{0.5(8)}{3} = \frac{4}{3} \approx 1.33$ [4 marks]