When we used the sample space {Rain, No Rain}, 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.
Step-by-step solution
Idea: A sample space must list every possible outcome exactly once, at the level of detail the question needs. If the question asks about something the sample space cannot show, the sample space is too rough.
- Rough sample space: S = {Rain, No Rain}, n(S) = 2. It is enough for “Should I carry an umbrella?” but it cannot tell drizzle from a downpour.½ mark
- Detailed sample space: the outcome “Rain” is split into Drizzle, Light Rain and Heavy Rain, giving S = {No Rain, Drizzle, Light Rain, Heavy Rain}, n(S) = 4. Now a question such as “Will the match be stopped by heavy rain?” can be answered.½ mark
- Both are proper sample spaces: each covers every possibility, and no outcome is listed twice or overlaps another (every day is exactly one of the four).½ mark
- Same idea elsewhere: for one roll of a die, {Even, Odd} is enough to ask “is it even?”, but {1, 2, 3, 4, 5, 6} is needed to ask “is it greater than 4?”. So we choose the sample space to suit the problem being studied.½ mark
Check: Test a sample space by asking: is every possible result in the list, exactly once, and can the event in the question be written as a set of these outcomes? If not, make it more detailed.
Answer to write in the exam
Whether it rains: S = {Rain, No Rain}, n(S) = 2
How much it rains: S = {No Rain, Drizzle, Light Rain, Heavy Rain}, n(S) = 4
Each lists every possible outcome exactly once.
∴ The sample space must be as detailed as the question requires.
Common mistakes that cost marks
- Writing {Rain, Drizzle, Light Rain, Heavy Rain}: “Rain” overlaps with the other three, so the same day would be counted twice. Keep outcomes that do not overlap.
- Leaving out an outcome, e.g. {Drizzle, Light Rain, Heavy Rain} with no “No Rain”. The sample space must include every possibility.
- Assuming the four weather outcomes are equally likely and giving each 14. A sample space just lists outcomes; it does not make them equally likely.
How this can come in the exam
Which of the following is a correct sample space for the result of a football match for one team, detailed enough to answer “Did the team avoid losing?”
- {Win}
- {Win, Lose}
- {Win, Draw, Lose}
- {Win, Draw, Lose, Win}
Show answer
(C) {Win, Draw, Lose}
A match can be won, drawn or lost; “avoid losing” = {Win, Draw}, so the draw must be a separate outcome, and nothing is listed twice.
For the marks a student scores in a test out of 10 (whole numbers only), write (a) a sample space that only shows whether the student passes (pass mark 4), (b) a sample space detailed enough to find whether the student scored more than 7.
Show answer
(a) {Pass, Fail}. (b) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}: 11 outcomes, from which “more than 7” = {8, 9, 10}.Try one yourself
A traffic light is observed at a random moment. Write a sample space with 2 outcomes and one with 3 outcomes.
Show answer
2 outcomes: {Stop (red), Not stop}. 3 outcomes: {Red, Amber, Green}.
More questions like this
- When a single 6-sided die is rolled, what is the total number of possible outcomes in the sample space?
- For the following experiments write down the sample space S.
- In a village fair, there are 3 popular snacks available: Samosa, Pakora, and Bhaji. For drinks, villagers can choose either Chai or Lassi.
- Experiment: Toss a fair coin two times.
- Can you calculate the probability of getting one head and one tail?