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Form 5 Probability Practice

Form 5 · 80 practice questions with instant marking and worked solutions.

Probability is an important part of the Form 5 mathematics curriculum. This topic develops sample spaces, equally likely outcomes, complements and checking. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Probability has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 80 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Probability; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 12 of them are DSE Paper 1 style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Probability set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Probability should also connect with the rest of mathematics. Number and algebra supply calculation tools; measurement and geometry give spatial meaning; data handling asks what a result means. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 80. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    In Probability, calculate $2/7$. Choose the correct answer.
  2. 2 Basic
    In Probability, calculate $3/8$. Give the numerical value only.

  3. 3 Basic
    In Probability, calculate $4/9$. Show the essential working and give the numerical value.

  4. 4 Basic
    In Probability, the value of $1/10$ is 1/5. Is this statement true?
  5. 5 Basic
    A bag has 3 red and 6 blue balls. Two are drawn at random without replacement. Find the probability that both are red (as a fraction in lowest terms). Choose the correct answer.
  6. 6 Basic
    A shooter hits the target with probability $\frac{1}{4}$ each time, independently. He shoots 3 times. Find the probability of at least one hit (as a fraction in lowest terms). Give the answer only.

  7. 7 Basic
    Of 31 students, 15 are in the basketball team and 11 of those are also in the swimming team. A basketball player is chosen at random. Find the probability that the student is also a swimmer (as a fraction in lowest terms). Show the essential working and give the answer.

  8. 8 Basic
    $A$ and $B$ are independent with $P(A)=\frac{3}{5}$ and $P(B)=\frac{1}{5}$. $P(A\cup B)$ is 4/5. Is this true?
  9. 9 Basic
    Roll a fair six-sided die. Find the probability that the result is even or greater than $4$.
  10. 10 Basic
    A fair spinner has $8$ equal-area sectors, of which $3$ are yellow. Find the probability of landing on a non-yellow sector as a simplified fraction.

  11. 11 Basic
    Roll two independent fair six-sided dice. Find the probability that their sum is $7$, as a simplified fraction.

  12. 12 Basic
    Events $A,B$ are mutually exclusive, with $P(A)=1/3$ and $P(B)=1/4$. Is the statement "$A,B$ are independent" true?
  13. 13 Basic
    A bag has 3 red and 4 blue balls. Two are drawn at random without replacement. Find the probability that they are of different colours (as a fraction in lowest terms). Choose the correct answer.
  14. 14 Basic
    A shooter hits the target with probability $\frac{1}{3}$ each time, independently. He shoots 4 times. Find the probability of at least one hit (as a fraction in lowest terms). Give the answer only.

  15. 15 Basic
    Of 38 students, 20 are in the basketball team and 3 of those are also in the swimming team. A basketball player is chosen at random. Find the probability that the student is also a swimmer (as a fraction in lowest terms). Show the essential working and give the answer.

  16. 16 Basic
    $A$ and $B$ are independent with $P(A)=\frac{3}{5}$ and $P(B)=\frac{1}{2}$. $P(A\cup B)$ is 11/10. Is this true?
  17. 17 Standard
    In Probability, calculate $2/6$. Choose the correct answer.
  18. 18 Standard
    In Probability, calculate $3/7$. Give the numerical value only.

  19. 19 Standard
    In Probability, calculate $4/8$. Show the essential working and give the numerical value.

  20. 20 Standard
    In Probability, the value of $1/9$ is 1/9. Is this statement true?
  21. 21 Standard
    4 books are chosen from 11 different books. Find the number of selections. Choose the correct answer.
  22. 22 Standard
    A bag has 7 red and 2 blue balls. Two are drawn at random without replacement. Find the probability that both are red (as a fraction in lowest terms). Give the answer only.

  23. 23 Standard
    A shooter hits the target with probability $\frac{2}{5}$ each time, independently. He shoots 4 times. Find the probability of at least one hit (as a fraction in lowest terms). Show the essential working and give the answer.

  24. 24 Standard DSE long question
    A bag has 6 red and 3 blue balls. Two are drawn without replacement. Answer each part; give the answer only.
    1. (a) Find the probability that the first is red (as a fraction in lowest terms).
    2. (b) Find the probability that both are red (as a fraction in lowest terms).
    3. (c) Find the probability of at least one blue ball (as a fraction in lowest terms).
  25. 25 Standard
    Toss a fair coin $3$ times independently. Find the probability of exactly two heads.
  26. 26 Standard
    Given $P(A)=2/5$, $P(B)=7/20$ and $P(A\cap B)=3/20$, find $P(A\cup B)$ as a simplified fraction.

  27. 27 Standard
    A bag contains $4$ red and $3$ blue balls. Draw one at random, replace it and mix before drawing again. Find the probability of red followed by blue as a simplified fraction.

  28. 28 Standard DSE long question
    A class has $40$ students: $24$ are in the music group, $18$ in the sports group, and $10$ in both. Select one student uniformly at random.
    1. (a) Find the probability that the student is in at least one group, as a simplified fraction.
    2. (b) Find the probability that the student is in neither group, as a simplified fraction.
    3. (c) Given that the student is in music, find the probability of also being in sports, as a simplified fraction.
  29. 29 Standard
    3 books are chosen from 10 different books. Find the number of selections. Choose the correct answer.
  30. 30 Standard
    A bag has 4 red and 6 blue balls. Two are drawn at random without replacement. Find the probability that both are red (as a fraction in lowest terms). Give the answer only.

  31. 31 Standard
    A shooter hits the target with probability $\frac{1}{2}$ each time, independently. He shoots 2 times. Find the probability of at least one hit (as a fraction in lowest terms). Show the essential working and give the answer.

  32. 32 Standard DSE long question
    A bag has 5 red and 5 blue balls. Two are drawn without replacement. Answer each part; give the answer only.
    1. (a) Find the probability that the first is red (as a fraction in lowest terms).
    2. (b) Find the probability that both are red (as a fraction in lowest terms).
    3. (c) Find the probability of at least one blue ball (as a fraction in lowest terms).
  33. 33 Advanced
    In Probability, calculate $2/10$. Choose the correct answer.
  34. 34 Advanced
    In Probability, calculate $3/6$. Give the numerical value only.

  35. 35 Advanced
    In Probability, calculate $4/7$. Show the essential working and give the numerical value.

  36. 36 Advanced
    In Probability, the value of $1/8$ is 1/4. Is this statement true?
  37. 37 Advanced
    Of 35 students, 19 are in the basketball team and 4 of those are also in the swimming team. A basketball player is chosen at random. Find the probability that the student is also a swimmer (as a fraction in lowest terms). Choose the correct answer.
  38. 38 Advanced
    $A$ and $B$ are independent with $P(A)=\frac{3}{5}$ and $P(B)=\frac{3}{5}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Give the answer only.

  39. 39 Advanced
    3 books are chosen from 12 different books. The number of selections is 220. Is this true?
  40. 40 Advanced DSE long question
    A group of 4 is chosen from 7 boys and 3 girls. Answer each part; give the answer only.
    1. (a) Find the total number of ways.
    2. (b) Find the number of ways with exactly 1 girl.
    3. (c) Find the number of ways with at least one girl.
  41. 41 Advanced
    Select an integer uniformly from $1$ to $30$. Given that it is a multiple of $3$, find the probability that it is also even.
  42. 42 Advanced
    A bag contains $5$ red and $4$ blue balls. Draw two at random without replacement. Find the probability that their colours differ, as a simplified fraction.

  43. 43 Advanced
    Given $P(A)=1/2$, $P(B)=1/3$ and $P(A\cap B)=1/6$, is the statement "$P(A\mid B)=1/2$, whereas $P(B\mid A)=1/3$" true?
  44. 44 Advanced DSE long question
    A fair coin is tossed twice independently. Let H denote heads and T denote tails.
    1. (a) Find the probability of at least one head, as a simplified fraction.
    2. (b) Given at least one head, find the probability of two heads, as a simplified fraction.
    3. (c) Given at least one head, find the probability of heads on the first toss, as a simplified fraction.
  45. 45 Advanced
    Of 51 students, 23 are in the basketball team and 7 of those are also in the swimming team. A basketball player is chosen at random. Find the probability that the student is also a swimmer (as a fraction in lowest terms). Choose the correct answer.
  46. 46 Advanced
    $A$ and $B$ are independent with $P(A)=\frac{3}{5}$ and $P(B)=\frac{2}{5}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Give the answer only.

  47. 47 Advanced
    5 books are chosen from 12 different books. The number of selections is 792. Is this true?
  48. 48 Advanced DSE long question
    A group of 3 is chosen from 5 boys and 5 girls. Answer each part; give the answer only.
    1. (a) Find the total number of ways.
    2. (b) Find the number of ways with exactly 2 girls.
    3. (c) Find the number of ways with at least one girl.
  49. 49 Challenging
    In Probability, calculate $2/9$. Choose the correct answer.
  50. 50 Challenging
    In Probability, calculate $3/10$. Give the numerical value only.

  51. 51 Challenging
    In Probability, calculate $4/6$. Show the essential working and give the numerical value.

  52. 52 Challenging
    In Probability, the value of $1/7$ is 1/7. Is this statement true?
  53. 53 Challenging
    A bag has 4 red and 3 blue balls. Two are drawn at random without replacement. Find the probability that both are red (as a fraction in lowest terms). Choose the correct answer.
  54. 54 Challenging
    A shooter hits the target with probability $\frac{1}{2}$ each time, independently. He shoots 4 times. Find the probability of at least one hit (as a fraction in lowest terms). Show the essential working and give the answer.

  55. 55 Challenging
    Of 39 students, 10 are in the basketball team and 4 of those are also in the swimming team. A basketball player is chosen at random. The probability that the student is also a swimmer is 4/39. Is this true?
  56. 56 Challenging DSE long question
    A bag has 5 red and 4 blue balls. Two are drawn without replacement. Answer each part; give the answer only.
    1. (a) Find the probability that the first is red (as a fraction in lowest terms).
    2. (b) Find the probability that both are red (as a fraction in lowest terms).
    3. (c) Find the probability of at least one blue ball (as a fraction in lowest terms).
  57. 57 Challenging
    The probability of rain on a day is $1/4$. Wa is late with probability $2/5$ if it rains and $1/10$ otherwise. Find the probability that Wa is late that day.
  58. 58 Challenging
    Events $A,B$ are independent, with $P(A)=2/5$ and $P(A\cup B)=7/10$. Find $P(B)$ as a simplified fraction.

  59. 59 Challenging
    Each attempt succeeds with probability $3/5$, and two attempts are independent. Is the statement "The probability of exactly one success is $6/5$" true?
  60. 60 Challenging DSE long question
    A bag contains $4$ red and $3$ blue distinguishable balls. Draw $3$ simultaneously, with every three-ball subset equally likely.
    1. (a) Find the probability of exactly two red balls, as a simplified fraction.
    2. (b) Find the probability that all three have the same colour, as a simplified fraction.
    3. (c) Find the probability of at least two red balls, as a simplified fraction.
  61. 61 Challenging
    A bag has 6 red and 2 blue balls. Two are drawn at random without replacement. Find the probability that both are red (as a fraction in lowest terms). Choose the correct answer.
  62. 62 Challenging
    A shooter hits the target with probability $\frac{1}{5}$ each time, independently. He shoots 4 times. Find the probability of at least one hit (as a fraction in lowest terms). Show the essential working and give the answer.

  63. 63 Challenging
    Of 41 students, 21 are in the basketball team and 15 of those are also in the swimming team. A basketball player is chosen at random. The probability that the student is also a swimmer is 15/41. Is this true?
  64. 64 Challenging DSE long question
    A bag has 7 red and 4 blue balls. Two are drawn without replacement. Answer each part; give the answer only.
    1. (a) Find the probability that the first is red (as a fraction in lowest terms).
    2. (b) Find the probability that both are red (as a fraction in lowest terms).
    3. (c) Find the probability of at least one blue ball (as a fraction in lowest terms).
  65. 65 Elite
    In Probability, calculate $2/8$. Choose the correct answer.
  66. 66 Elite
    In Probability, calculate $3/9$. Give the numerical value only.

  67. 67 Elite
    In Probability, calculate $4/10$. Show the essential working and give the numerical value.

  68. 68 Elite
    In Probability, the value of $1/6$ is 1/3. Is this statement true?
  69. 69 Elite
    $A$ and $B$ are independent with $P(A)=\frac{3}{10}$ and $P(B)=\frac{3}{5}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Give the answer only.

  70. 70 Elite
    2 books are chosen from 11 different books. Find the number of selections. Show the essential working and give the answer.

  71. 71 Elite
    A bag has 5 red and 6 blue balls. Two are drawn at random without replacement. The probability that both are red is 2/11. Is this true?
  72. 72 Elite DSE long question
    A group of 3 is chosen from 4 boys and 5 girls. Answer each part; give the answer only.
    1. (a) Find the total number of ways.
    2. (b) Find the number of ways with exactly 1 girl.
    3. (c) Find the number of ways with at least one girl.
  73. 73 Elite
    Roll a fair six-sided die twice independently. Given at least one $6$, find the probability that only the second roll is $6$, as a simplified fraction.

  74. 74 Elite
    Select $3$ students uniformly from $4$ boys and $3$ girls. Given that the group contains at least one girl, find the probability that it contains a particular girl, Mei, as a simplified fraction.

  75. 75 Elite
    A bag has $2$ red and $3$ blue balls. Draw two without replacement. Given at least one red ball, is the statement "The conditional probability of two red balls is $1/7$" true?
  76. 76 Elite DSE long question
    Bag A has $3$ red and $1$ blue ball; bag B has $1$ red and $3$ blue balls. Choose a bag using a fair coin, then draw at random from that bag. A second draw, if made, uses the same bag without replacing the first ball.
    1. (a) Find the probability that the first ball is red, as a simplified fraction.
    2. (b) Given a red first ball, find the probability that bag A was chosen, as a simplified fraction.
    3. (c) Find the probability of two consecutive red balls, as a simplified fraction.
  77. 77 Elite
    $A$ and $B$ are independent with $P(A)=\frac{2}{5}$ and $P(B)=\frac{3}{5}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Give the answer only.

  78. 78 Elite
    2 books are chosen from 10 different books. Find the number of selections. Show the essential working and give the answer.

  79. 79 Elite
    A bag has 3 red and 5 blue balls. Two are drawn at random without replacement. The probability that both are red is 3/28. Is this true?
  80. 80 Elite DSE long question
    A group of 4 is chosen from 4 boys and 6 girls. Answer each part; give the answer only.
    1. (a) Find the total number of ways.
    2. (b) Find the number of ways with exactly 2 girls.
    3. (c) Find the number of ways with at least one girl.

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Probability

This topic needs particular care with sample spaces, equally likely outcomes, complements and checking. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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