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Secondary 2 Mathematics Classroom | Chapter 11: Probability, Sample Spaces and Combined Events | G2/G3

SECONDARY 2 MATHEMATICS CLASSROOM · CHAPTER 11 · PROBABILITY · SAMPLE SPACES · COMBINED EVENTS · G2/G3

Probability: When Possible Outcomes Become a Structured Space

Probability is not guessing. It is counting outcomes carefully, defining what is possible, and comparing favourable outcomes with the whole sample space.

Chapter 10 described observed data. Chapter 11 turns from what happened to what could happen. The same discipline remains: define the denominator honestly, compare like with like, and make no claim stronger than the structure allows.

Classroom rule: define the experiment → list or model the sample space → decide whether outcomes are equally likely → identify the event → count favourable outcomes → calculate → use complements or multiplicative structure when helpful → check the result lies between 0 and 1.

Level boundary. The shared G2/G3 core here is simple probability, sample spaces, experimental probability, complements and straightforward combined events. More complex dependence, formal conditional probability or advanced combinatorics should be treated as later-course material unless already introduced in the learner’s current sequence.

Official reference: MOE G2 and G3 Mathematics Syllabuses.

Featured Answer: What Is a Sample Space?

A sample space is the complete set of possible outcomes for a random experiment. A probability calculation is only as reliable as the sample space underneath it.

How to Use This Classroom

  1. State the random experiment clearly.
  2. List all possible outcomes without omission or duplication.
  3. Check whether the outcomes are equally likely.
  4. Define the event precisely.
  5. Count favourable outcomes only after the full sample space is stable.
  6. Use probability between 0 and 1 inclusive.
  7. Use complements when “not” is easier to count.
  8. For combined events, distinguish with-replacement from without-replacement situations.
  9. Use tree diagrams when sequential structure matters.
  10. Compare experimental results with theoretical expectations carefully.

1. Probability Measures Likelihood

A probability of 0 means impossible. A probability of 1 means certain. Values between 0 and 1 represent degrees of likelihood.

2. Equally Likely Outcomes Give the Basic Fraction Rule

P(event)=number of favourable outcomes / total number of equally likely outcomes

3. Teacher Model 1: One Die

For a fair six-sided die, the sample space is {1,2,3,4,5,6}. The event “roll an even number” is {2,4,6}, so P(even)=3/6=1/2.

4. The Denominator Comes From the Whole Sample Space

Do not divide by the number of favourable outcomes or by the number of outcome types you happen to notice.

5. Teacher Model 2: One Card From a Small Set

A bag contains cards labelled 1,2,3,4,5,6,7,8. Probability of selecting a number greater than 5 is 3/8.

6. Sample Spaces Can Be Listed Systematically

For two coin tosses, the ordered sample space is {HH, HT, TH, TT}. HT and TH are different outcomes because the order differs.

7. Teacher Model 3: Two Coins

Event “exactly one head”={HT,TH}. Therefore P(exactly one head)=2/4=1/2.

8. Tables Help With Two-Stage Sample Spaces

If one spinner has outcomes A,B and another has 1,2,3, the combined ordered outcomes are A1,A2,A3,B1,B2,B3.

9. Tree Diagrams Show Sequential Structure

Each branch represents a possible next outcome. Multiplying branch probabilities gives the probability of a complete path when the branch probabilities are correctly defined.

10. Teacher Model 4: Coin Then Die

P(H and then 6)=1/2×1/6=1/12.

11. “And” Often Uses Path Multiplication

For independent stages, multiply along a path. But do not use multiplication mechanically when the second-stage probability changes because an item was removed.

12. “Or” Often Requires Combining Distinct Favourable Outcomes

If the event can happen through separate non-overlapping paths, add those path probabilities.

13. Teacher Model 5: Exactly One Head in Two Tosses

P(HT)+P(TH)=1/4+1/4=1/2.

14. Complementary Events Add to 1

P(not A)=1−P(A)

15. Teacher Model 6: At Least One Head

In two fair coin tosses, “at least one head” is easier through the complement “no heads”. P(TT)=1/4, so P(at least one head)=1−1/4=3/4.

16. Experimental Probability Comes From Observed Frequency

experimental probability = observed frequency / number of trials

17. Teacher Model 7: Experimental Result

A spinner lands on red 37 times in 100 trials. Experimental probability=37/100=0.37.

18. Experimental and Theoretical Probabilities Need Not Match Exactly

Random variation causes observed frequencies to fluctuate. With many trials, results may stabilise closer to the theoretical model if the model is appropriate.

19. Expected Frequency Uses Probability as a Rate

expected frequency = probability × number of trials

If P(red)=0.3 and there are 200 trials, expected frequency=60.

20. Without Replacement Changes the Sample Space

If an item is removed and not returned, both the numerator and denominator for later draws can change.

21. Teacher Model 8: Two Red Without Replacement

A bag has 3 red and 2 blue counters. Draw two without replacement.

P(red then red)=3/5×2/4=3/10.

22. With Replacement Keeps the Stage Probabilities Stable

If the first counter is returned before the second draw, P(red then red)=3/5×3/5=9/25.

23. Ordered Outcomes Matter in Sequential Events

Red then blue and blue then red are different paths even if the final colour counts are the same.

24. Teacher Model 9: One Red and One Blue

Using the same 3-red,2-blue bag without replacement:

P(RB)=3/5×2/4=3/10.

P(BR)=2/5×3/4=3/10.

P(one of each)=3/5.

25. Do Not Assume Outcomes Are Equally Likely

A spinner with unequal sectors does not give each colour equal probability merely because each colour appears once.

26. Teacher Model 10: Unequal Spinner

If red occupies half the spinner and blue and green each occupy one quarter, P(red)=1/2, not 1/3.

27. Misconception Clinic: Probability Can Be Greater Than 1

Repair: probabilities lie from 0 to 1 inclusive.

28. Misconception Clinic: Two Coins Have Three Outcomes

Repair: HH, HT, TH and TT are four ordered outcomes.

29. Misconception Clinic: HT and TH Are the Same Outcome

Repair: sequence matters when stages are ordered.

30. Misconception Clinic: Experimental Probability Must Equal Theoretical Probability

Repair: random variation means finite experiments need not match theory exactly.

31. Misconception Clinic: Without Replacement Uses the Same Denominator Again

Repair: one item has been removed, so the total number remaining changes.

32. Misconception Clinic: Every Named Outcome Is Equally Likely

Repair: equal likelihood must come from the physical or mathematical model, not from the labels.

33. Guided Practice A: Simple Probability

  1. Fair die: P(number greater than 4).
  2. Cards 1–10: P(odd).
  3. Cards 1–10: P(not odd).
Solutions

2/6=1/3. 5/10=1/2. 1/2.

34. Guided Practice B: Two-Stage Outcomes

  1. Two fair coins: P(two heads).
  2. Two fair coins: P(exactly one tail).
  3. Two fair coins: P(at least one head).
Solutions

1/4. 1/2. 3/4.

35. Guided Practice C: Without Replacement

A bag has 4 red and 3 blue counters. Two are drawn without replacement.

  1. Find P(RR).
  2. Find P(BB).
  3. Find P(one of each).
Solutions

P(RR)=4/7×3/6=2/7. P(BB)=3/7×2/6=1/7. P(one of each)=4/7×3/6+3/7×4/6=4/7.

36. Guided Practice D: Experimental Probability

A coin shows heads 58 times in 100 tosses.

  1. Find experimental P(H).
  2. Compare with theoretical P(H) for a fair coin.
  3. Explain why the two values need not match exactly.
Solutions

0.58. Theoretical value 0.5. Random variation in a finite number of trials can produce a different observed proportion.

37. Challenge Practice: Complement Route

A fair die is rolled twice. Find the probability of getting at least one 6.

Worked solution

P(no 6)=5/6×5/6=25/36. Therefore P(at least one 6)=1−25/36=11/36.

38. Assessment Method: Build the Sample Space Before the Fraction

Most probability errors begin with an incomplete or distorted set of possible outcomes.

39. Assessment Method: Check Equal Likelihood

The basic favourable/total rule assumes equally likely elementary outcomes.

40. Assessment Method: Use Complements Strategically

“At least one” is often easier as 1−P(none).

41. Assessment Method: Recalculate After Removal

Without replacement, update both the number of favourable items and the total remaining.

42. Oral Classroom Check

  1. What is a sample space?
  2. What range can probability take?
  3. Why are HT and TH different?
  4. When can favourable/total be used directly?
  5. What is a complement?
  6. Why is “at least one” often solved using a complement?
  7. What is experimental probability?
  8. Why can experiment and theory differ?
  9. What changes without replacement?
  10. Why are named outcomes not automatically equally likely?

43. Exit Ticket

  1. Fair die: P(odd).
  2. Fair die: P(not odd).
  3. Two fair coins: P(HH).
  4. Two fair coins: P(exactly one head).
  5. Two fair coins: P(at least one head).
  6. A spinner lands red 42 times in 120 trials. Find experimental P(red).
  7. A bag has 3 red and 2 blue counters. Find P(RR) without replacement.
  8. Explain why the denominator changes on the second draw.
  9. State the complement rule.
  10. Explain why a probability of 1.2 is impossible.
Exit-ticket solutions

1/2. 1/2. 1/4. 1/2. 3/4. 42/120=0.35. 3/5×2/4=3/10. One counter has been removed, leaving four total. P(not A)=1−P(A). Probability cannot exceed certainty, represented by 1.

44. Homework: Retrieval, Variation and Transfer

  • list four complete sample spaces;
  • solve six simple probability questions;
  • solve two complement questions;
  • solve two tree-diagram questions;
  • solve two without-replacement questions;
  • compare one experimental result with a theoretical model;
  • write one explanation of why equal labels do not imply equal likelihood.

45. The Seven-Day Return Cycle

  1. Day 0: sample spaces and simple events.
  2. Day 1: complements and experimental probability.
  3. Day 3: mixed tree/without-replacement work.
  4. Day 7: changed exit ticket with one “at least one” problem.

46. The Full Probability Routine

define experiment → build sample space → check equal likelihood → define event → count or multiply paths → combine paths or use complement → update after removal → verify 0≤P≤1.

47. Connect Back to Chapter 10

Return to Secondary 2 Chapter 10: Statistics, Data Representation and Misleading Graphs when experimental frequency, denominator control or fair comparison is unstable. Statistics describes observed outcomes; probability models possible outcomes.

48. Specialist Companions

49. Why This Chapter Matters for Chapter 12

Chapter 12 brings the whole Secondary 2 year together. Probability contributes sample-space discipline, denominator control and uncertainty reasoning to mixed-topic problem solving. The year-end synthesis will require students to decide which mathematical structure owns a problem before calculating.

50. Ready for Chapter 12?

  • build complete sample spaces;
  • calculate simple theoretical probabilities;
  • use experimental probability;
  • use complement reasoning;
  • combine sequential events correctly;
  • distinguish with and without replacement;
  • update denominators after removal;
  • check equal likelihood before using favourable/total;
  • verify every probability lies between 0 and 1.

If one item is weak, return to the smallest section that owns it and solve a changed example. When the route is stable, continue to Chapter 12: Whole-Year Synthesis, Modelling, Communication and Secondary 3 Handover.