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DSE M1 Binomial, Geometric and Poisson Distributions Practice

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

Binomial, Geometric and Poisson Distributions is an important part of the DSE M1 mathematics curriculum. This topic develops the conditions, probabilities, means and variances of binomial, geometric and Poisson distributions. 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 Binomial, Geometric and Poisson Distributions 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 Binomial, Geometric and Poisson Distributions; 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 Binomial, Geometric and Poisson Distributions 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.

Binomial, Geometric and Poisson Distributions 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.

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  1. 1 Basic
    $X\sim B(5,\frac{1}{3})$. Find $P(X=3)$ (as a fraction in lowest terms). Choose the correct answer.
  2. 2 Basic
    $X\sim B(10,\frac{1}{4})$. Find $E(X)$ (as a fraction in lowest terms). Give the answer only.

  3. 3 Basic
    Each shot scores with probability $\frac{1}{2}$, independently. $X$ is the number of shots up to and including the first score. Find $P(X=3)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  4. 4 Basic
    Customers arrive at a Poisson rate of $\frac{5}{2}$ per hour. $Y$ is the number arriving in 4 hours. $\text{Var}(Y)$ is 5/2. Is this true?
  5. 5 Basic
    $X\sim B(5,\frac{1}{4})$. Find $P(X=3)$ (as a fraction in lowest terms). Choose the correct answer.
  6. 6 Basic
    $X\sim B(25,\frac{2}{5})$. Find $\text{Var}(X)$. Give the answer only.

  7. 7 Basic
    Each shot scores with probability $\frac{1}{5}$, independently. $X$ is the number of shots up to and including the first score. Find $P(X=4)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  8. 8 Basic
    Customers arrive at a Poisson rate of $\frac{5}{2}$ per hour. $Y$ is the number arriving in 4 hours. $E(Y)$ is 5/2. Is this true?
  9. 9 Basic
    $X\sim B(6,\frac{1}{2})$. Find $P(X=5)$ (as a fraction in lowest terms). Choose the correct answer.
  10. 10 Basic
    $X\sim B(21,\frac{1}{4})$. Find $\text{Var}(X)$ (as a fraction in lowest terms). Give the answer only.

  11. 11 Basic
    Each shot scores with probability $\frac{1}{4}$, independently. $X$ is the number of shots up to and including the first score. Find $P(X>5)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  12. 12 Basic
    Assume independent arrival counts in disjoint intervals and a constant arrival rate. Customers arrive at a Poisson rate of $5$ per hour. $Y$ is the number arriving in 4 hours. $E(Y)$ is 5. Is this true?
  13. 13 Standard
    $X$ follows a Poisson distribution with mean $5$. Find the value of $c$ when $P(X=3)$ is written as $ce^{-5}$ (as a fraction in lowest terms). Choose the correct answer.
  14. 14 Standard
    $X\sim B(4,\frac{1}{2})$. Find $P(X=1)$ (as a fraction in lowest terms). Give the answer only.

  15. 15 Standard
    $X\sim B(51,\frac{2}{5})$. Find $E(X)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  16. 16 Standard DSE long question
    Each test succeeds with probability $\frac{2}{5}$, independently; 7 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$ (as a fraction in lowest terms).
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  17. 17 Standard
    $X$ follows a Poisson distribution with mean $\frac{3}{2}$. Find the value of $c$ when $P(X=2)$ is written as $ce^{-\frac{3}{2}}$ (as a fraction in lowest terms). Choose the correct answer.
  18. 18 Standard
    $X\sim B(6,\frac{1}{2})$. Find $P(X=0)$ (as a fraction in lowest terms). Give the answer only.

  19. 19 Standard
    $X\sim B(28,\frac{1}{4})$. Find $\text{Var}(X)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  20. 20 Standard DSE long question
    Each test succeeds with probability $\frac{1}{5}$, independently; 5 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$.
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  21. 21 Standard
    $X$ follows a Poisson distribution with mean $\frac{1}{2}$. Find the value of $c$ when $P(X\le 1)$ is written as $ce^{-\frac{1}{2}}$ (as a fraction in lowest terms). Choose the correct answer.
  22. 22 Standard
    $X\sim B(3,\frac{1}{2})$. Find $P(X=0)$ (as a fraction in lowest terms). Give the answer only.

  23. 23 Standard
    $X\sim B(13,\frac{2}{5})$. Find $E(X)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  24. 24 Standard DSE long question
    Each test succeeds with probability $\frac{1}{3}$, independently; 6 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$.
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  25. 25 Advanced
    Each shot scores with probability $\frac{1}{3}$, independently. $X$ is the number of shots up to and including the first score. Find $P(X=4)$ (as a fraction in lowest terms). Choose the correct answer.
  26. 26 Advanced
    Customers arrive at a Poisson rate of $\frac{3}{2}$ per hour. $Y$ is the number arriving in 2 hours. Find $E(Y)$. Give the answer only.

  27. 27 Advanced
    $X$ follows a Poisson distribution with mean $4$. The value of $c$ when $P(X=1)$ is written as $ce^{-4}$ is 4. Is this true?
  28. 28 Advanced DSE long question
    Each test succeeds with probability $\frac{1}{2}$, independently; 3 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$ (as a fraction in lowest terms).
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  29. 29 Advanced
    Each shot scores with probability $\frac{2}{5}$, independently. $X$ is the number of shots up to and including the first score. Find $P(X=5)$ (as a fraction in lowest terms). Choose the correct answer.
  30. 30 Advanced
    Customers arrive at a Poisson rate of $2$ per hour. $Y$ is the number arriving in 2 hours. Find $\text{Var}(Y)$. Give the answer only.

  31. 31 Advanced
    $X$ follows a Poisson distribution with mean $4$. The value of $c$ when $P(X=2)$ is written as $ce^{-4}$ is 8. Is this true?
  32. 32 Advanced DSE long question
    Each test succeeds with probability $\frac{1}{3}$, independently; 5 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$ (as a fraction in lowest terms).
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  33. 33 Advanced
    Each shot scores with probability $\frac{2}{5}$, independently. $X$ is the number of shots up to and including the first score. Find $P(X=2)$ (as a fraction in lowest terms). Choose the correct answer.
  34. 34 Advanced
    Assume independent arrival counts in disjoint intervals and a constant arrival rate. Customers arrive at a Poisson rate of $2$ per hour. $Y$ is the number arriving in 3 hours. Find $\text{Var}(Y)$. Give the answer only.

  35. 35 Advanced
    $X$ follows a Poisson distribution with mean $5$. The value of $c$ when $P(X=0)$ is written as $ce^{-5}$ is 1. Is this true?
  36. 36 Advanced DSE long question
    Each test succeeds with probability $\frac{1}{2}$, independently; 6 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$.
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  37. 37 Challenging
    $X\sim B(4,\frac{2}{5})$. Find $P(X=3)$ (as a fraction in lowest terms). Choose the correct answer.
  38. 38 Challenging
    $X\sim B(36,\frac{1}{2})$. Find $\text{Var}(X)$. Show the essential working and give the answer.

  39. 39 Challenging
    Customers arrive at a Poisson rate of $5$ per hour. $Y$ is the number arriving in 4 hours. $\text{Var}(Y)$ is 5. Is this true?
  40. 40 Challenging DSE long question
    Each test succeeds with probability $\frac{1}{2}$, independently; 5 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$ (as a fraction in lowest terms).
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  41. 41 Challenging
    $X\sim B(5,\frac{1}{3})$. Find $P(X=0)$ (as a fraction in lowest terms). Choose the correct answer.
  42. 42 Challenging
    $X\sim B(40,\frac{2}{5})$. Find $\text{Var}(X)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  43. 43 Challenging
    Each shot scores with probability $\frac{1}{3}$, independently. $X$ is the number of shots up to and including the first score. $P(X=3)$ is 8/81. Is this true?
  44. 44 Challenging DSE long question
    Each test succeeds with probability $\frac{1}{4}$, independently; 3 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$ (as a fraction in lowest terms).
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  45. 45 Challenging
    $X\sim B(5,\frac{1}{4})$. Find $P(X=0)$ (as a fraction in lowest terms). Choose the correct answer.
  46. 46 Challenging
    $X\sim B(46,\frac{3}{10})$. Find $\text{Var}(X)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  47. 47 Challenging
    Each shot scores with probability $\frac{1}{4}$, independently. $X$ is the number of shots up to and including the first score. $P(X=5)$ is 243/4096. Is this true?
  48. 48 Challenging DSE long question
    Each test succeeds with probability $\frac{1}{2}$, independently; 7 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$ (as a fraction in lowest terms).
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  49. 49 Elite
    Customers arrive at a Poisson rate of $3$ per hour. $Y$ is the number arriving in 4 hours. Find $\text{Var}(Y)$. Give the answer only.

  50. 50 Elite
    $X$ follows a Poisson distribution with mean $2$. Find the value of $c$ when $P(X\le 1)$ is written as $ce^{-2}$. Show the essential working and give the answer.

  51. 51 Elite
    $X\sim B(6,\frac{1}{3})$. $P(X=4)$ is 20/243. Is this true?
  52. 52 Elite DSE long question
    Each test succeeds with probability $\frac{1}{4}$, independently; 4 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$.
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  53. 53 Elite
    Customers arrive at a Poisson rate of $\frac{5}{2}$ per hour. $Y$ is the number arriving in 4 hours. Find $\text{Var}(Y)$. Give the answer only.

  54. 54 Elite
    $X$ follows a Poisson distribution with mean $5$. Find the value of $c$ when $P(X=2)$ is written as $ce^{-5}$ (as a fraction in lowest terms). Show the essential working and give the answer.

  55. 55 Elite
    $X\sim B(3,\frac{1}{4})$. $P(X=2)$ is 9/64. Is this true?
  56. 56 Elite DSE long question
    Each test succeeds with probability $\frac{2}{5}$, independently; 3 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$ (as a fraction in lowest terms).
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (as a fraction in lowest terms).
  57. 57 Elite
    Assume independent arrival counts in disjoint intervals and a constant arrival rate. Customers arrive at a Poisson rate of $4$ per hour. $Y$ is the number arriving in 2 hours. Find $\text{Var}(Y)$. Give the answer only.

  58. 58 Elite
    $X$ follows a Poisson distribution with mean $4$. Find the value of $c$ when $P(X\le 1)$ is written as $ce^{-4}$. Show the essential working and give the answer.

  59. 59 Elite
    $X\sim B(5,\frac{1}{4})$. $P(X=5)$ is 1/1024. Is this true?
  60. 60 Elite DSE long question
    Each test succeeds with probability $\frac{1}{3}$, independently; 4 tests are done. $X$ is the number of successes. Answer each part; give the answer only.
    1. (a) Find $E(X)$ (as a fraction in lowest terms).
    2. (b) Find $\text{Var}(X)$ (as a fraction in lowest terms).
    3. (c) Find $P(X=0)$ (as a fraction in lowest terms).
    4. (d) Find $P(X\ge 1)$ (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 Binomial, Geometric and Poisson Distributions

This topic needs particular care with the conditions, probabilities, means and variances of binomial, geometric and Poisson distributions. 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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