Secondary 5 Sample Space and Events Worksheet

All 3 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Probability –

   A carnival stall runs the following operation. A spinner with four congruent sectors marked $2$, $3$, $5$ and $7$ is spun once, then a balanced four-sided die numbered $1$ to $4$ is rolled once. The result recorded is the ordered pair (spinner number, die number).

   1. Explain why this operation is a random experiment.
   2. Give the sample space $\Omega$ and its cardinality.
   3. The stall pays a prize when the product of the two numbers is even. Find the probability of winning a prize.
   4. Find the probability that the product exceeds $20$.
2. Q2Probability –

   A city is studying a new bike lane. Consider the experiment “choose one cyclist at random on a given weekday and observe whether that cyclist uses the new lane”.

   Write three short statements about this situation, one that would give a theoretical probability, one an experimental probability and one a subjective probability. Explain in one line what makes each statement belong to its type.

3. Q3Probability –

   A survey of 500 households asked about two composting programs. It found that 320 households own a compost bin, 250 own a rain barrel, and 90 own neither.

   1. Describe, in words, the event complementary to “the household owns neither”.
   2. Give the probability of that complementary event.
   3. How many households own both? Give the probability.
   4. Find the probability that a household chosen at random owns a compost bin but no rain barrel.

The 2 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

What makes an experiment random — three conditions, three sentences (Q1a)

"Explain why this operation is a random experiment" is not a warm-up; it is a definition question, and it is marked as three separate claims. An operation is a random experiment when all three hold:

1. It can be repeated as often as you like under the same conditions. 2. All its possible results are known before it is carried out. 3. The result of any one trial cannot be predicted with certainty.

The second and third conditions sound contradictory until you separate them: you know the list of what could happen, you just cannot say which entry of that list will happen. A complete answer applies each of the three to the actual situation in the question rather than reciting the definition — spinning the same spinner again is the same conditions, the sixteen possible pairs are known in advance, and no single spin is predictable.

Writing the sample space of a two-stage experiment (Q1b)

When an experiment has two stages and the result recorded is an ordered pair, the sample space is every possible first result matched with every possible second result. Write it in set-builder form and then show enough of the list for it to be unambiguous:

Its cardinality comes from the multiplication principle — four choices at the first stage, each followed by four at the second, gives sixteen outcomes. The word ordered matters: the pair (2, 3) is a different outcome from (3, 2), and the two stages here are different objects anyway.

Counting an event without counting anything twice (Q1c, Q1d)

"The product is even" is the classic place to lose a mark, because the natural first thought — add the outcomes where the first number is even to the outcomes where the second number is even — counts twice every outcome where both are even.

Theoretical, experimental, subjective — and writing them yourself (Q2)

Q2 turns the usual question inside out: instead of classifying three given statements, you have to compose one of each about the same situation, and justify each in a line. That is harder, and it is the version that shows whether the distinction is understood.

Theoretical — the value comes from a model of equiprobable cases, counted before any observation. Your statement must therefore contain a model: some collection of cases and a reason they are interchangeable. Experimental — the value is a relative frequency read off repeated real trials, so your statement must contain a count and a number of trials. Subjective — the value is somebody's judgment, so your statement must name whose judgment it is, and it can be neither counted nor measured.

The one-line justifications are where the marks sit, and the shortest honest summary is worth memorising: a theoretical probability comes from a model, an experimental one from data, and a subjective one from an opinion. A useful test on your own three sentences — ask what more data would do to each. Only the experimental value would move.

Two overlapping groups: one identity does all four parts (Q3)

Q3 gives you how many households own each of two things and how many own neither, and asks for the overlap and for one group without the other. Everything comes from:

The subtraction is there for exactly the reason it was needed above — the households owning both were counted once in each of the first two totals, so one copy has to come back out. Rearranged, it hands you the overlap directly.

The route through the question: "owns neither" is the complement of "owns at least one", so subtracting it from the total gives n(A ∪ B). The identity then gives n(A ∩ B). And "owns a compost bin but no rain barrel" is the households owning a bin, minus those owning both — a subtraction, not a new count.

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Getting the most out of it

Draw the grid, or the two-way table

A two-stage experiment with a manageable number of outcomes belongs in a rectangle: first result down the side, second across the top, the recorded value in each cell. Two overlapping groups belong in a two-by-two table with a totals row and column. Both turn a reasoning question into a reading question, and both make double counting visible instead of invisible.

Say the event in words before you write the fraction

"At least one is even", "neither owns one", "the complement of that" — write the sentence, then decide whether to count it directly, split it into non-overlapping cases, or count its complement. Most errors on this sheet are committed in the translation step, not in the arithmetic.

Justify equiprobability once, out loud

One line at the start — congruent sectors, balanced die, so the outcomes are equally likely — licenses every count that follows and is itself worth marks. On a question where the outcomes are not equally likely, noticing that is the entire question.

Common questions