Probability of Combined Events · Form 4
Probability of Combined Events: Key Terms
The key Probability of Combined Events terms you need for SPM Mathematics, defined plainly in English, Malay and Chinese.
- Probability A measure between 0 and 1 of how likely an event is to happen.
- Independent events Events where one outcome does not affect the probability of the other.
- Mutually exclusive events Events that cannot both happen at the same time.
- Tree diagram A branching diagram that lays out the outcomes and probabilities of combined events.
- Sample space The set of all possible outcomes of an experiment, written S.
- Complementary event The event that A does not happen; its probability is 1 − P(A).
- Favourable outcome An outcome that counts as the event you are finding the probability of.
- Dependent events Events where one happening changes the probability of the other.
Term pairs students confuse
- Independent vs dependent events, the difference is whether the first outcome changes the second: independent (with replacement) keeps every probability the same; dependent (without replacement) lowers the totals on the next draw.
- Mutually exclusive vs independent, the difference is what they claim: mutually exclusive means the two events cannot both happen, so P(A∩B) = 0; independent means one does not affect the other, so P(A∩B) = P(A) × P(B). They are not the same idea.
- Complementary vs mutually exclusive events, the difference is completeness: complementary events are mutually exclusive and together cover everything, so their probabilities add to 1 (A and A'); merely mutually exclusive events need not fill the whole sample space.
- Sample space vs favourable outcome, the difference is scope: the sample space is every possible outcome, while a favourable outcome is one that matches the event you want; probability is favourable outcomes ÷ sample space.
- 'And' vs 'or' the difference is the operation: 'A and B' (∩) means both, so you multiply along a branch; 'A or B' (∪) means either, so you add, subtracting any overlap.
How these words appear in real SPM questions
- 'Two marbles are drawn one after another without replacement' signals dependent events, reduce the total on the second draw.
- 'A card is drawn, noted and replaced' or 'with replacement' signals independent events, the probabilities stay the same.
- 'Find the probability that at least one …' is a cue for the complement, 1 − P(none).
- 'Both … and …' or 'first … then …' tells you to multiply along a single branch of the tree.
- 'Either … or …' asks for the addition law; check whether the two events can happen together before deciding to subtract P(A∩B).
- Notation counts as wording: P(A∩B) means 'A and B' (multiply for independent events) and P(A∪B) means 'A or B' (add, then subtract the overlap).
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Can two events be both mutually exclusive and independent?
In practice, no, not for ordinary events. If they are mutually exclusive then P(A∩B) = 0, but independence needs P(A∩B) = P(A) × P(B).
Those agree only when one event has probability zero. So treat 'mutually exclusive' and 'independent' as different, even opposite, descriptions of a pair of events.
Is a complementary event the same as a mutually exclusive one?
They overlap but are not the same. Complementary events A and A' are mutually exclusive and also cover the whole sample space, so their probabilities add to exactly 1.
Two events can be mutually exclusive without being complementary, rolling a 2 or a 5 excludes each other but does not fill every outcome.
What is the difference between the sample space and an event?
The sample space is the full list of every possible outcome of the experiment. An event is a chosen subset of that list, the outcomes you are interested in.
Probability compares the two: the number of favourable outcomes in the event divided by the number of outcomes in the sample space.