Probability measures uncertainty; it does not remove it. A probability of 0.8 does not promise that an event will occur on the next trial. It says the event is assigned a high chance under the stated model. Probability becomes reliable when the sample space is defined clearly, outcomes are counted correctly, and assumptions such as equally likely outcomes are justified rather than guessed.
This Secondary 2 Mathematics Learning Guide develops probability from possible outcomes, sample spaces, complements and repeated trials. It also introduces simple combined-event structures as a bridge for students whose current school course includes them, while keeping the core reasoning grounded in lower-secondary chance models.
Secondary Mathematics Hub: S1–S4 Capability Map · Secondary 2 Learning Guide, Batch 3, Guide 4. Companion guides cover trigonometry, mensuration, and statistics.
Course boundary. The current MOE G2/G3 Mathematics syllabus includes probability as a measure of chance and probability of single events, including listing all possible outcomes in simple chance situations. Exact progression differs by school. This guide treats single-event probability and sample spaces as the core, with simple two-stage and combined-event examples clearly functioning as a bridge rather than assuming every Secondary 2 class has reached the same depth.
Navigate: Probability scale · Sample spaces · Equally likely outcomes · Complements · Experimental probability · Combined events · Practice and answers · Teaching and transfer.
1. Probability lives between impossible and certain
A probability is a number from 0 to 1 inclusive. Probability 0 means impossible under the model. Probability 1 means certain. Values closer to 1 represent greater chance; values closer to 0 represent smaller chance.
Probabilities may also be written as fractions, decimals or percentages. For example, 3/4 = 0.75 = 75%. Changing representation does not change the underlying chance.
Worked example 1: locate probability on the scale
An event with probability 0.02 is possible but unlikely. An event with probability 0.5 is neither guaranteed nor impossible; in a symmetric two-outcome model it may represent equal chance. An event with probability 1.2 is invalid because probability cannot exceed 1.
Probability describes the model, not a feeling
Saying an outcome feels likely is not enough. A mathematical probability needs a defined mechanism, observed data or a justified model. If a spinner has unequal sectors, counting colours without considering sector size may be wrong.
Probability begins by asking: what outcomes are possible, and what makes them equally or unequally likely?
2. A sample space lists all possible outcomes under the model
The sample space is the complete set of possible outcomes. For one ordinary six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. For one coin toss, it is {H, T}.
The word complete matters. If one possible outcome is omitted, every probability based on the sample space may be distorted.
Worked example 2: single die event
For a fair six-sided die, find the probability of an even number. Favourable outcomes are {2, 4, 6}, so there are 3 favourable outcomes out of 6 equally likely outcomes. Probability = 3/6 = 1/2.
Worked example 3: one spinner
A spinner has eight equal sectors: three red, two blue and three green. Probability of blue = 2/8 = 1/4. Probability of red or green = 6/8 = 3/4.
The equal-sector condition justifies counting sectors. If sector areas were unequal, simply counting labels would not necessarily represent chance.
Worked example 4: outcomes versus events
Rolling a 4 is one outcome. Rolling an even number is an event containing three outcomes: 2, 4 and 6. An event is therefore a set of outcomes satisfying a stated condition.
3. Favourable over total works only when outcomes are equally likely
The familiar formula probability = favourable outcomes / total outcomes depends on the listed outcomes having equal chance. For a fair die, the six face results are equally likely. For a bag containing individual identical tokens, each token may be equally likely to be selected if mixing and selection are fair.
But suppose a wheel has one red sector covering half the circle and three blue sectors dividing the other half. Counting sector labels gives one red against three blue, but probability of red is 1/2, not 1/4, because the sectors are not equal in size.
Worked example 5: identical tokens in a bag
A bag contains 5 red, 3 blue and 2 yellow identical tokens. One token is selected at random. There are 10 equally likely individual-token outcomes. Probability of blue = 3/10. Probability of not yellow = 8/10 = 4/5.
Counting colours as three equally likely outcomes would be wrong because there are different numbers of tokens of each colour.
Worked example 6: number condition
A card is selected from cards numbered 1 to 12, one card of each number. Probability of selecting a multiple of 3: favourable outcomes are 3, 6, 9, 12, so probability = 4/12 = 1/3.
Probability of selecting a prime number: favourable outcomes 2, 3, 5, 7, 11, so probability = 5/12.
4. The complement is everything in the sample space outside the event
If event A has probability P(A), then the probability that A does not occur is 1 − P(A). The event and its complement together cover the complete sample space and do not overlap.
Worked example 7: use a complement directly
If probability of rain under a stated model is 0.35, then probability of no rain is 1 − 0.35 = 0.65.
This is not a weather forecast claim; it is a mathematical relationship between an event and its complement under whatever model supplied 0.35.
Worked example 8: at least one through complement
As an extension to two-stage reasoning, suppose a fair coin is tossed twice. The sample space is HH, HT, TH, TT. Probability of at least one head = 3/4. An efficient alternative is 1 − probability of no heads = 1 − 1/4 = 3/4.
Complement reasoning becomes powerful when the unwanted case is easier to count than many favourable cases.
5. Experimental probability comes from observed relative frequency
If an event occurs 37 times in 100 trials, its experimental probability or relative frequency is 37/100 = 0.37. This describes the observed data, not necessarily the exact theoretical probability of the mechanism.
With more trials under stable conditions, experimental proportions often become more stable around the model’s underlying probability. But short runs can fluctuate substantially.
Worked example 9: compare experiment with model
A fair coin is tossed 40 times and shows 23 heads. Experimental probability of heads = 23/40 = 0.575. The theoretical model gives 0.5. The difference does not by itself prove the coin is unfair; random variation occurs in finite samples.
If many large experiments consistently depart strongly from the theoretical model, that would motivate checking the assumptions or mechanism. Probability and statistics meet here: data can test whether a model seems plausible.
Worked example 10: use experimental probability for an estimate
A machine-like classroom simulation produces event A on 84 of 300 trials. Estimated probability = 84/300 = 0.28. If the same conditions are assumed for another 500 trials, an estimated number of A outcomes is 0.28 × 500 = 140.
This is an expectation based on the observed rate, not a guarantee that exactly 140 will occur.
6. Bridge: combined events need a complete two-stage sample space
This section is a bridge where the current school course permits it. When an experiment has two stages, outcomes can be listed systematically using an ordered list, table or tree diagram. The main danger is missing combinations or counting them with the wrong weights.
Worked example 11: two fair coin tosses
The sample space is {HH, HT, TH, TT}. These four ordered outcomes are equally likely. Probability of exactly one head = 2/4 = 1/2. Notice HT and TH are distinct outcomes because order records which toss produced the head.
Worked example 12: coin and die
A fair coin and fair six-sided die are used. There are 2 × 6 = 12 equally likely ordered outcomes: H1 through H6 and T1 through T6. Probability of head and an even number = outcomes H2, H4, H6, so 3/12 = 1/4.
The multiplication 2 × 6 counts combinations because each coin outcome can pair with each die outcome. It does not by itself calculate probability unless equal likelihood is justified.
Worked example 13: with replacement versus without replacement
A bag contains 2 red and 1 blue token. If a token is drawn, replaced and drawn again, the composition returns to 2 red and 1 blue before the second draw. If it is not replaced, the second-stage probabilities depend on the first result because the bag composition changes.
This distinction should be read from the wording before any tree diagram is built. Replacement preserves the original composition; no replacement changes it.
Worked example 14: simple without-replacement probability
From a bag with 2 red and 1 blue token, draw two without replacement. Probability of red then blue = (2/3)(1/2) = 1/3. Probability of blue then red = (1/3)(2/2) = 1/3.
Therefore probability of one red and one blue in either order = 1/3 + 1/3 = 2/3. The two routes are mutually exclusive ordered paths: both cannot occur in the same two-draw experiment.
7. Independence is a relationship between events, not a synonym for unrelated stories
In a simple model, two events are independent when knowing that one occurred does not change the probability of the other. A coin toss and a separate fair die roll are commonly modelled as independent.
Two draws without replacement from one bag are generally dependent because the first draw changes the bag. Avoid memorising multiply means independent without checking the structure of the experiment.
A tree diagram is a probability map
Branch probabilities leaving the same node should sum to 1 because they represent all possible next outcomes from that state. Multiplying along a path combines the successive conditions on that route. Adding mutually exclusive path probabilities combines alternative routes to the same requested event.
This bridge is useful even before formal combined-event rules are assessed because it forces the learner to represent what changes from stage to stage.
8. Common probability errors are usually sample-space errors first
- Probability greater than 1: repair scale meaning.
- Favourable/total used with unequal outcomes: repair equal-likelihood assumption.
- Coin tossed twice listed as H and T only: repair multi-stage sample space.
- HT and TH treated as identical when order matters: repair outcome representation.
- Replacement ignored: repair changing-state model.
- Experimental 0.57 treated as proof theoretical probability is 0.57: repair distinction between observation and model.
- At least one counted incompletely: consider complement reasoning.
- Percent chance interpreted as a guarantee: repair uncertainty meaning.
9. Mixed practice: define the experiment before counting
Questions 1–6. 1. State the sample space for one fair six-sided die. 2. Find probability of a number greater than 4. 3. Find probability of a factor of 6. 4. A bag contains 4 red, 3 blue and 1 green identical token. Find probability of blue. 5. Find probability of not green. 6. Explain why probability 1.08 is impossible.
Questions 7–12. 7. Cards numbered 1 to 10 are equally likely. Find probability of an even number. 8. Find probability of a prime number. 9. If P(A) = 0.27, find P(not A). 10. An event occurs 42 times in 120 trials. Find experimental probability. 11. Use it to estimate occurrences in 500 similar trials. 12. Explain why that estimated count is not guaranteed.
Bridge questions 13–18. 13. List the sample space for two fair coin tosses. 14. Find probability of two heads. 15. Find probability of at least one head. 16. A fair coin and fair die are used. Find probability of tail and a number greater than 4. 17. A bag contains 2 red and 2 blue tokens. Two are drawn without replacement. Find probability of red then red. 18. Explain why the second red probability changes after the first red has been removed.
Explained answers: questions 1–6
1. {1,2,3,4,5,6}. 2. {5,6}: 2/6 = 1/3. 3. {1,2,3,6}: 4/6 = 2/3. 4. 3/8. 5. 7/8. 6. Probability cannot exceed 1.
Explained answers: questions 7–12
7. Five even numbers out of 10, so 1/2. 8. Primes 2,3,5,7: 4/10 = 2/5. 9. 0.73. 10. 42/120 = 0.35. 11. 0.35 × 500 = 175. 12. Random outcomes fluctuate; 175 is an expectation based on the observed rate.
Explained answers: bridge questions 13–18
13. {HH, HT, TH, TT}. 14. 1/4. 15. 3/4. 16. Tail gives 1/2 and die greater than 4 gives 2/6; combined probability = 1/6 under independence.
17. First red 2/4, then red 1/3, giving 1/6. 18. After one red is removed, only one red remains among three tokens; the state of the bag has changed.
10. Teaching sequence: enumerate before applying rules
Begin with physical or imagined experiments small enough to list completely: one die, one spinner, numbered cards. Ask students to write every possible outcome before calculating. Then distinguish outcomes from events and ask which assumption makes the outcomes equally likely.
Next compare theoretical and experimental probability. Run or imagine repeated trials and discuss why short-run frequencies move. Introduce complement as a complete-partition idea. Only after the single-stage sample space is secure should two-stage tables or trees be used as a bridge.
Questions parents and tutors can ask
What exactly is one trial? What are all possible outcomes? Are they equally likely? Which outcomes satisfy the event? What is the complement? Does replacement occur? Does the first stage change the second? Is this an observed frequency or a theoretical probability? Can the answer be larger than 1?
11. The transfer test: chance language versus mathematical structure
A fictional game uses a fair six-sided die. A player wins on 5 or 6. The theoretical probability of winning one round is 2/6 = 1/3. Over 300 rounds, a long-run estimate is about 100 wins, but exactly 100 is not promised.
Suppose 300 actual rounds produce 112 wins. The experimental probability is 112/300 ≈ 0.373. That observed result is above 1/3, but one experiment does not rewrite the theoretical model automatically.
If the die mechanism is genuinely fair, variation is expected. If repeated large datasets show persistent systematic departure, the assumption of fairness becomes a question worth investigating. Probability supplies the model; data supply evidence about how well the model matches the world.
Define the experiment. List the complete sample space. Check equal likelihood. Count the event correctly. Distinguish theoretical chance from observed frequency. Let the structure decide whether a multi-stage bridge is appropriate.
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