Combined-event probability means finding the chance that two or more events happen together. This section has four lessons and a practice set, and belongs to the wider SPM Mathematics guide.
What does this section cover?
Start with drawing sample spaces systematically, so every outcome is listed once. Then learn to distinguish independent and dependent events, which decides whether the second probability changes.
Next is using tree diagrams with and without replacement. The last lesson covers combining mutually exclusive and non-exclusive events. The practice set mixes all four.
One bag, four different rules
Take this example. A bag holds 3 red and 2 blue cards. Two cards are drawn one after another.
| Question | What to notice | Rule |
|---|---|---|
| List all outcomes of drawing two cards | Every pair must be listed | Sample space |
| Card returned after the first draw | Second draw unchanged | Independent: 3/5 × 3/5 for two reds |
| Card kept after the first draw | Second draw changes | Dependent: 3/5 × 2/4 = 3/10 for two reds |
| Both cards red or both blue | Cannot happen together | Mutually exclusive: add, 3/10 + 1/10 = 2/5 |
The bag never changed. The wording changed, and with it the rule.
Who should start where?
If you forget outcomes when listing, start with sample spaces. If you multiply when you should add, begin with mutually exclusive events.
If your second probability never changes when it should, start with independent and dependent events. If tree diagrams feel slow, use the tree diagram lesson.
For a teacher to go through your reasoning question by question, see online one-to-one Mathematics tuition.