Q1.
When we used the sample space {Rain, No Rain} in Example 1, we focused only on whether it will rain or not. However, if we want to include different amounts of rainfall like drizzle, light rain or heavy rain, we need to expand the sample space to {No Rain, Drizzle, Light Rain, Heavy Rain} so that it better matches the level of detail required for the question. It is important to ensure the sample space is detailed enough to suit the specific problem being studied.
Answer
The point is that a sample space is not fixed by the weather — it is chosen by the question you are asking. Both lists below are correct sample spaces for tomorrow; they answer different questions.
| Question being studied | Suitable sample space | n(S) |
|---|---|---|
| Should I carry an umbrella? | S = {Rain, No Rain} | 2 |
| Will the school ground be fit for the match? | S = {No Rain, Drizzle, Light Rain, Heavy Rain} | 4 |
Whichever list you pick, it must obey the two rules of Section 7.3.1:
- Nothing left out — every possible result must appear, or some trial will have no outcome to belong to.
- Nothing listed twice — the entries must not overlap, or one trial would count as two outcomes.
Why it happens: the finer list is built by splitting the single outcome "Rain" into Drizzle, Light Rain and Heavy Rain. Splitting keeps the list complete and non-overlapping, so it is still a valid sample space — but notice what it does not do. It does not make the four outcomes equally likely. Writing P(Heavy Rain) = 1/4 just because n(S) = 4 would be wrong; these probabilities have to come from rainfall records, exactly as on page 157.
Tip: "detailed enough" cuts both ways. Too coarse and the sample space cannot express the event you care about — {Rain, No Rain} has no way to say "heavy rain". Too fine (rainfall to the nearest millimetre) and you will never collect enough data to estimate any of the probabilities. Match the detail to the decision.