O-Level E-Math and SEC G3 Mathematics K310
S2: Probability
Define the sample space, use complements or combined events carefully, and check that final probabilities are plausible.
Core notes
Probability measures chance on a scale from to . Define the sample space, represent combined stages systematically, and distinguish mutually exclusive events from independent events.
Probability and single events
For equally likely outcomes, probability is favourable outcomes divided by total outcomes. The complement of event has probability . Listing outcomes is safest when the sample space is small.
Combined events and diagrams
Possibility diagrams display ordered pairs of outcomes, while tree diagrams display successive stages. Multiply along a complete path and add probabilities of disjoint paths that satisfy the required event.
Addition and multiplication rules
For mutually exclusive events, there is no overlap, so add their probabilities. For independent events, one event does not change the other, so multiply their probabilities for both to occur. If events overlap or depend on earlier outcomes, adjust the model rather than applying a rule mechanically.
Formulae and methods
| Relationship | Use |
|---|---|
| Use the complement of an event. | |
| Add mutually exclusive events. | |
| Multiply independent events. |
Worked examples
Example 1: A fair die is rolled twice. Find the probability that both results are even.
- One roll is even with probability .
- The rolls are independent.
- Multiply: .
Answer: The probability is .
Example 2: A bag has red and blue counters. Two are drawn without replacement. Find the probability of one counter of each colour.
- Use two disjoint orders: red then blue, or blue then red.
- Calculate .
- Simplify .
Answer: The probability is .
Chapter checkpoint
- List or model the complete sample space before counting favourable outcomes.
- Decide whether events are mutually exclusive, independent, conditional or neither.
- Check that the final probability lies between and and that branch probabilities sum correctly.
Exam traps and retrieval check
Avoid these traps
- Omitting possible outcomes from the sample space.
- Treating draws without replacement as independent.
- Adding probabilities for events that overlap without subtracting the overlap.
Check from memory
What must all probabilities lie between?
and , inclusive.
What does mutually exclusive mean?
The events cannot occur together.
What does independence mean?
The occurrence of one event does not change the probability of the other.
Official K310 coverage
This chapter maps to 4 official outcomes: S2.1, S2.2, S2.3, S2.4. Check the official 2027 K310 syllabus for exact assessable wording and paper details.