Probability of Combined Events · Form 4
What is a sample space?
The sample space is the complete list of every possible outcome of an experiment, usually written as a set S. Without the full sample space you cannot work out any probability from it.
The set of everything that can happen
A sample space, S, contains every outcome an experiment can produce, with no repeats. Rolling one fair die gives S = {1, 2, 3, 4, 5, 6}, so n(S) = 6, where n(S) is the number of outcomes.
Each single outcome is called a sample point, and any event you talk about later is just a part of this set.
It is the denominator of every probability
Probability of an event = (number of favourable outcomes) ÷ n(S), so the sample space is what you divide by every time. For combined events the space gets bigger: tossing two coins gives S = {HH, HT, TH, TT}, so n(S) = 4, not 2.
Getting n(S) wrong throws off every answer that follows, which is why listing the space carefully comes first.
Spotting it, and the usual slip
The sample space is the whole list; an event is only the outcomes that satisfy a condition, so don't mix them up. A common slip with combined events is forgetting that order can create separate outcomes, HT and TH are two different points, even though both are 'one head, one tail'.
Another is stopping the list early, so double-check that n(S) matches what you expect: two dice give 6 × 6 = 36 outcomes, not 12.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How is a sample space different from an event?
The sample space, S, lists every possible outcome of an experiment, while an event is a specific subset of outcomes you're interested in, such as "getting a sum of 7." You always list the sample space first, then pick out which outcomes belong to your event before calculating probability.
What's the best way to list the sample space for two combined events, like tossing two coins or rolling two dice?
A table or tree diagram is the safest method, it lays out every combination systematically so no outcome is missed or repeated. For two dice, a table with 6 rows and 6 columns gives all 36 outcomes clearly, which is far more reliable than trying to list them by hand.
What mistake causes students to lose marks when writing out a sample space?
The most common mistake is missing outcomes or counting the same outcome twice, especially when order matters, for example, treating "red then blue" and "blue then red" as identical when they should be counted separately. Always check the total count of your sample space matches what the method predicts, like 6 × 6 = 36 for two dice.