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DSE M1 Further Probability Practice

DSE M1 · 60 practice questions with instant marking and worked solutions.

Further Probability is an important part of the DSE M1 mathematics curriculum. This topic develops conditional probability, independent events, the law of total probability and Bayes' theorem. 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 Further 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 60 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 Further 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 Extended Part 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 Further 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.

Further Probability is closely linked with the rest of the Extended Part: algebraic and trigonometric skills underpin calculus, and calculus in turn supports probability distributions and statistical inference. 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 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

Start practising

  1. 1 Basic
    Of 56 students, 28 are in the basketball team and 25 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.
  2. 2 Basic
    Machine A makes $\frac{1}{2}$ of the output with defect rate $\frac{1}{25}$; machine B makes the rest with defect rate $\frac{1}{20}$. A randomly chosen item is defective. Find the probability that it came from machine A (as a fraction in lowest terms). Give the answer only.

  3. 3 Basic
    $A$ and $B$ are independent with $P(A)=\frac{3}{10}$ and $P(B)=\frac{1}{5}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  4. 4 Basic
    A disease has prevalence $\frac{3}{5}$. A test is positive with probability $\frac{1}{2}$ for a person with the disease and $\frac{1}{5}$ for a person without it. The probability that a randomly chosen person tests positive is 3/10. Is this true?
  5. 5 Basic
    Of 43 students, 19 are in the basketball team and 14 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.
  6. 6 Basic
    Machine A makes $\frac{3}{5}$ of the output with defect rate $\frac{1}{25}$; machine B makes the rest with defect rate $\frac{1}{20}$. A randomly chosen item is defective. Find the probability that it came from machine A (as a fraction in lowest terms). Give the answer only.

  7. 7 Basic
    $A$ and $B$ are independent with $P(A)=\frac{3}{10}$ and $P(B)=\frac{3}{10}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  8. 8 Basic
    A disease has prevalence $\frac{1}{4}$. A test is positive with probability $\frac{3}{5}$ for a person with the disease and $\frac{1}{5}$ for a person without it. The probability that a randomly chosen person tests positive is 3/20. Is this true?
  9. 9 Basic
    Of 26 students, 13 are in the basketball team and 6 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.
  10. 10 Basic
    Machine A makes $\frac{7}{10}$ of the output with defect rate $\frac{1}{25}$; machine B makes the rest with defect rate $\frac{1}{20}$. A randomly chosen item is defective. Find the probability that it came from machine A (as a fraction in lowest terms). Give the answer only.

  11. 11 Basic
    $A$ and $B$ are independent with $P(A)=\frac{1}{5}$ and $P(B)=\frac{2}{5}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  12. 12 Basic
    A disease has prevalence $\frac{1}{2}$. A test is positive with probability $\frac{4}{5}$ for a person with the disease and $\frac{1}{10}$ for a person without it. The probability that a randomly chosen person tests positive is 2/5. Is this true?
  13. 13 Standard
    $P(A)=\frac{1}{2}$, $P(B)=\frac{3}{5}$ and $P(A\cup B)=\frac{9}{10}$. Find $P(A|B)$ (as a fraction in lowest terms). Choose the correct answer.
  14. 14 Standard
    Of 38 students, 13 are in the basketball team and 10 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). Give the answer only.

  15. 15 Standard
    Machine A makes $\frac{3}{5}$ of the output with defect rate $\frac{1}{10}$; machine B makes the rest with defect rate $\frac{1}{50}$. A randomly chosen item is defective. Find the probability that it came from machine A (as a fraction in lowest terms). Show the essential working and give the answer.

  16. 16 Standard DSE long question
    Machine A makes $\frac{3}{5}$ of the output (defect rate $\frac{1}{20}$); machine B makes the rest (defect rate $\frac{1}{50}$). Answer each part; give the answer only.
    1. (a) Find the probability that a random item is defective (as a fraction in lowest terms).
    2. (b) Given it is defective, find the probability it came from A (as a fraction in lowest terms).
    3. (c) Find the probability that an item is not defective (as a fraction in lowest terms).
  17. 17 Standard
    $P(A)=\frac{1}{2}$, $P(B)=\frac{3}{5}$ and $P(A\cup B)=\frac{7}{10}$. Find $P(A|B)$ (as a fraction in lowest terms). Choose the correct answer.
  18. 18 Standard
    Of 39 students, 21 are in the basketball team and 10 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). Give the answer only.

  19. 19 Standard
    Machine A makes $\frac{2}{5}$ of the output with defect rate $\frac{1}{10}$; machine B makes the rest with defect rate $\frac{1}{50}$. A randomly chosen item is defective. Find the probability that it came from machine A (as a fraction in lowest terms). Show the essential working and give the answer.

  20. 20 Standard DSE long question
    Machine A makes $\frac{2}{5}$ of the output (defect rate $\frac{1}{20}$); machine B makes the rest (defect rate $\frac{1}{25}$). Answer each part; give the answer only.
    1. (a) Find the probability that a random item is defective (as a fraction in lowest terms).
    2. (b) Given it is defective, find the probability it came from A (as a fraction in lowest terms).
    3. (c) Find the probability that an item is not defective (as a fraction in lowest terms).
  21. 21 Standard
    $P(A)=\frac{1}{2}$, $P(B)=\frac{2}{5}$ and $P(A\cup B)=\frac{7}{10}$. Find $P(A|B)$ (as a fraction in lowest terms). Choose the correct answer.
  22. 22 Standard
    Of 27 students, 13 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). Give the answer only.

  23. 23 Standard
    Machine A makes $\frac{1}{2}$ of the output with defect rate $\frac{1}{10}$; machine B makes the rest with defect rate $\frac{1}{20}$. A randomly chosen item is defective. Find the probability that it came from machine A (as a fraction in lowest terms). Show the essential working and give the answer.

  24. 24 Standard DSE long question
    Machine A makes $\frac{3}{5}$ of the output (defect rate $\frac{1}{10}$); machine B makes the rest (defect rate $\frac{1}{50}$). Answer each part; give the answer only.
    1. (a) Find the probability that a random item is defective (as a fraction in lowest terms).
    2. (b) Given it is defective, find the probability it came from A (as a fraction in lowest terms).
    3. (c) Find the probability that an item is not defective (as a fraction in lowest terms).
  25. 25 Advanced
    $A$ and $B$ are independent with $P(A)=\frac{1}{2}$ and $P(B)=\frac{3}{10}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Choose the correct answer.
  26. 26 Advanced
    A disease has prevalence $\frac{3}{5}$. A test is positive with probability $\frac{4}{5}$ for a person with the disease and $\frac{3}{10}$ for a person without it. Find the probability that a randomly chosen person tests positive (as a fraction in lowest terms). Give the answer only.

  27. 27 Advanced
    $P(A)=\frac{2}{5}$, $P(B)=\frac{3}{10}$ and $P(A\cup B)=\frac{3}{5}$. $P(A|B)$ is 1/3. Is this true?
  28. 28 Advanced DSE long question
    A bag has 4 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).
  29. 29 Advanced
    $A$ and $B$ are independent with $P(A)=\frac{3}{10}$ and $P(B)=\frac{2}{5}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Choose the correct answer.
  30. 30 Advanced
    A disease has prevalence $\frac{1}{2}$. A test is positive with probability $\frac{3}{5}$ for a person with the disease and $\frac{1}{10}$ for a person without it. Find the probability that a randomly chosen person tests positive (as a fraction in lowest terms). Give the answer only.

  31. 31 Advanced
    $P(A)=\frac{3}{5}$, $P(B)=\frac{1}{2}$ and $P(A\cup B)=\frac{4}{5}$. $P(A|B)$ is 3/5. Is this true?
  32. 32 Advanced DSE long question
    A bag has 3 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
    $A$ and $B$ are independent with $P(A)=\frac{2}{5}$ and $P(B)=\frac{3}{10}$. Find $P(A\cup B)$ (as a fraction in lowest terms). Choose the correct answer.
  34. 34 Advanced
    A disease has prevalence $\frac{3}{10}$. A test is positive with probability $\frac{7}{10}$ for a person with the disease and $\frac{1}{10}$ for a person without it. Find the probability that a randomly chosen person tests positive (as a fraction in lowest terms). Give the answer only.

  35. 35 Advanced
    $P(A)=\frac{3}{10}$, $P(B)=\frac{3}{10}$ and $P(A\cup B)=\frac{2}{5}$. $P(A|B)$ is 2/3. Is this true?
  36. 36 Advanced DSE long question
    A bag has 6 red and 2 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).
  37. 37 Challenging
    Of 57 students, 29 are in the basketball team and 8 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 Challenging
    Machine A makes $\frac{7}{10}$ of the output with defect rate $\frac{1}{10}$; machine B makes the rest with defect rate $\frac{3}{100}$. A randomly chosen item is defective. Find the probability that it came from machine A (as a fraction in lowest terms). Show the essential working and give the answer.

  39. 39 Challenging
    A disease has prevalence $\frac{3}{5}$. A test is positive with probability $\frac{7}{10}$ for a person with the disease and $\frac{3}{10}$ for a person without it. The probability that a randomly chosen person tests positive is 21/50. Is this true?
  40. 40 Challenging DSE long question
    Machine A makes $\frac{1}{2}$ of the output (defect rate $\frac{1}{20}$); machine B makes the rest (defect rate $\frac{1}{25}$). Answer each part; give the answer only.
    1. (a) Find the probability that a random item is defective (as a fraction in lowest terms).
    2. (b) Given it is defective, find the probability it came from A (as a fraction in lowest terms).
    3. (c) Find the probability that an item is not defective (as a fraction in lowest terms).
  41. 41 Challenging
    Of 34 students, 17 are in the basketball team and 8 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.
  42. 42 Challenging
    Machine A makes $\frac{2}{5}$ of the output with defect rate $\frac{1}{25}$; machine B makes the rest with defect rate $\frac{3}{100}$. A randomly chosen item is defective. Find the probability that it came from machine A (as a fraction in lowest terms). Show the essential working and give the answer.

  43. 43 Challenging
    $A$ and $B$ are independent with $P(A)=\frac{3}{10}$ and $P(B)=\frac{1}{5}$. $P(A\cup B)$ is 1/2. Is this true?
  44. 44 Challenging DSE long question
    Machine A makes $\frac{1}{2}$ of the output (defect rate $\frac{1}{20}$); machine B makes the rest (defect rate $\frac{1}{50}$). Answer each part; give the answer only.
    1. (a) Find the probability that a random item is defective (as a fraction in lowest terms).
    2. (b) Given it is defective, find the probability it came from A (as a fraction in lowest terms).
    3. (c) Find the probability that an item is not defective (as a fraction in lowest terms).
  45. 45 Challenging
    Of 36 students, 12 are in the basketball team and 8 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 Challenging
    Machine A makes $\frac{2}{5}$ of the output with defect rate $\frac{1}{20}$; machine B makes the rest with defect rate $\frac{1}{50}$. A randomly chosen item is defective. Find the probability that it came from machine A (as a fraction in lowest terms). Show the essential working and give the answer.

  47. 47 Challenging
    $A$ and $B$ are independent with $P(A)=\frac{2}{5}$ and $P(B)=\frac{1}{2}$. $P(A\cup B)$ is 9/10. Is this true?
  48. 48 Challenging DSE long question
    Machine A makes $\frac{1}{2}$ of the output (defect rate $\frac{1}{10}$); machine B makes the rest (defect rate $\frac{1}{25}$). Answer each part; give the answer only.
    1. (a) Find the probability that a random item is defective (as a fraction in lowest terms).
    2. (b) Given it is defective, find the probability it came from A (as a fraction in lowest terms).
    3. (c) Find the probability that an item is not defective (as a fraction in lowest terms).
  49. 49 Elite
    A disease has prevalence $\frac{3}{5}$. A test is positive with probability $\frac{3}{5}$ for a person with the disease and $\frac{3}{10}$ for a person without it. Find the probability that a randomly chosen person tests positive (as a fraction in lowest terms). Give the answer only.

  50. 50 Elite
    $P(A)=\frac{3}{5}$, $P(B)=\frac{1}{2}$ and $P(A\cup B)=\frac{9}{10}$. Find $P(A|B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  51. 51 Elite
    Of 29 students, 11 are in the basketball team and 5 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 5/11. Is this true?
  52. 52 Elite DSE long question
    A bag has 4 red and 2 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).
  53. 53 Elite
    $P(A)=\frac{1}{2}$, $P(B)=\frac{3}{10}$ and $P(A\cup B)=\frac{3}{5}$. Find $P(A|B)$ (as a fraction in lowest terms). Give the answer only.

  54. 54 Elite
    $P(A)=\frac{2}{5}$, $P(B)=\frac{3}{10}$ and $P(A\cup B)=\frac{1}{2}$. Find $P(A|B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  55. 55 Elite
    Of 37 students, 27 are in the basketball team and 13 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 13/27. Is this true?
  56. 56 Elite DSE long question
    A bag has 3 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).
  57. 57 Elite
    A disease has prevalence $\frac{2}{5}$. A test is positive with probability $\frac{7}{10}$ for a person with the disease and $\frac{1}{5}$ for a person without it. Find the probability that a randomly chosen person tests positive (as a fraction in lowest terms). Give the answer only.

  58. 58 Elite
    $P(A)=\frac{1}{2}$, $P(B)=\frac{2}{5}$ and $P(A\cup B)=\frac{3}{5}$. Find $P(A|B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  59. 59 Elite
    Of 48 students, 28 are in the basketball team and 11 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 11/28. Is this true?
  60. 60 Elite DSE long question
    A bag has 7 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).

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

Common mistakes

Common mistakes in Further Probability

This topic needs particular care with conditional probability, independent events, the law of total probability and Bayes' theorem. 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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