Probability of Combined Events
Independent events
Events where one outcome does not affect the probability of the other.
| English | Independent events |
|---|---|
| Bahasa Melayu | Peristiwa tak bersandar |
| 中文 | 独立事件 |
How it is used
Tossing a fair coin then rolling a fair die are independent. P(heads and a 6) = P(heads) × P(6) = 1/2 × 1/6 = 1/12, because the coin result does not change the die's chances.
Where it shows up in SPM
Central to combined-events questions, especially those using the multiplication rule P(A and B) = P(A) × P(B). In Paper 2 it appears with tree diagrams for 'with replacement' problems, where an item is returned before the second draw so probabilities stay the same.
Don't confuse it with
Open the chapter: Probability of Combined Events →
Frequently asked questions
How do I know if events are independent in a question?
Look for signals like tossing separate coins, rolling different dice, or drawing 'with replacement'. If one action's result cannot affect the next, they are independent.
Two draws from the same bag are independent only when the first item is put back before the second draw.
Do I add or multiply probabilities for independent events?
For 'A and B both happening', multiply: P(A) × P(B). The word 'and' along a single path of a tree diagram signals multiplication.
You only add when combining separate paths that lead to the required outcome, which is the 'or' step.
Are independent and mutually exclusive the same thing?
No, they are almost opposites. Independent events can occur together and P(both) = P(A) × P(B) is usually not zero.
Mutually exclusive events cannot occur together, so P(both) = 0. Mixing them up is a common exam mistake, so read the situation carefully.