To list outcomes without gaps, hold one thing fixed, run through every option for the next thing, then move the fixed thing on. Order is the whole technique.
What is the small gap?
Students list outcomes as they occur to them. A random list always looks complete, because nothing in it reveals what is absent. A fixed order reveals gaps as soon as a pattern breaks.
How do you list, step by step?
- Decide what makes one outcome: a pair, a triple, a number.
- Fix the first item, and list every option for the second.
- Move the first item on and repeat.
- Count the list and compare with the product of the choices.
A scaffolded example
An original problem: a canteen meal set has one main (nasi lemak or mee goreng) and one drink (teh, kopi or sirap). List all possible sets and find the probability of a set with mee goreng.
Fix nasi lemak: nasi lemak with teh, kopi, sirap. Then fix mee goreng: mee goreng with teh, kopi, sirap.
| Teh | Kopi | Sirap | |
|---|---|---|---|
| Nasi lemak | NL-T | NL-K | NL-S |
| Mee goreng | MG-T | MG-K | MG-S |
The count is 2 × 3 = 6, matching the table. Three of the six have mee goreng, so the probability is 3/6 = 1/2, assuming each set is equally likely.
What does the common mistake look like?
A student asked about two coins writes HH, HT, TT, and says there are three outcomes, so P(one head) = 1/3. The list dropped TH. With the fixed-order method: first coin H gives HH, HT; first coin T gives TH, TT. There are 4, and P(one head) = 2/4 = 1/2.
The counting check also fails: 2 × 2 = 4, not 3.
Self-check
A fair die is rolled and a coin is tossed. List the outcomes where the die shows an even number and the coin shows heads, and find the probability.
Answer
The total outcomes are 6 × 2 = 12. Fix each die value: 1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T.
Even die and heads: 2H, 4H, 6H, which is 3 outcomes. Probability = 3/12 = 1/4.
Where does this show up in SPM?
In SPM Mathematics, listing outcomes underlies probability, sets and later the combined-events work. In Science, genetics crosses, such as Punnett squares, use the same two-way layout.
Next, see how listing connects to real trial results in comparing experimental and theoretical probability. The hub shows the complete sequence.
One-to-one Mathematics tuition starts with a one-hour trial class (from RM50).