121i - Identifying Sample Spaces

Lesson 121 — Identifying the Sample Space for Single-Stage Events

Strand: Probability | Descriptor: AC9M7P01 | Duration: 45 minutes

Equipment: dice, coins, a spinner or two, a pack of cards.

Learning Intentions

• To identify the sample space of a single-stage chance experiment.
• To list outcomes systematically and completely.

Success Criteria

I can:

1. Define a chance experiment, an outcome and the sample space.
2. List a sample space systematically, with nothing missing or repeated.
3. Distinguish an outcome from an event.
4. Count the outcomes favourable to a given event.

Warmup

(6 minutes — list them all, pairs)

For each, list every possible result.

1. Tossing a coin.
2. Rolling a standard die.
3. Spinning a spinner with four equal sectors labelled A, B, C, D.
4. Drawing one card from a pack and noting its suit.
5. Drawing one card and noting whether it is red or black.

Answers: 1. ; 2. ; 3. ; 4. ; 5. .

Notice Q4 and Q5: the same experiment has different sample spaces depending on what we record. The sample space is not a property of the equipment — it depends on the question asked.

Activities

Activity 1 — Explicit Instruction: the Vocabulary (14 min)

The three terms, defined precisely:

Term	Meaning	Example (rolling a die)
Chance experiment	A process with an uncertain result	Rolling the die
Outcome	One possible result	Rolling a
Sample space	The set of all possible outcomes
Event	A collection of outcomes we care about	”Rolling an even number”

The outcome/event distinction is the lesson’s core. An outcome is a single result; an event may contain several. “Rolling a ” is one outcome and also an event containing one outcome. “Rolling more than ” is an event containing two outcomes: .

Notation to establish: sample spaces and events are written in curly brackets, , with meaning “the number of outcomes in “.

Two requirements for a correct sample space:

1. Complete — every possible outcome is listed.
2. No repeats — each outcome appears exactly once.

I do — list systematically, not randomly. Drawing one letter from the word PROBABILITY:

Random attempt: P, R, O, B, A, B, I, L, I, T, Y — has repeats.

Systematic attempt, alphabetical: A, B, I, L, O, P, R, T, Y, so .

Emphasise: working alphabetically or numerically is what guarantees completeness. Random listing loses outcomes and duplicates others.

We do — list the sample space and the named event:

1. Rolling a die; event: “a factor of “.
2. Spinning a spinner with sectors –; event: “a multiple of “.
3. Drawing a letter from MATHEMATICS; event: “a vowel”.
4. Choosing a month at random; event: “a month with days”.

(Answers: 1. , event ; 2. , event ; 3. A, C, E, H, I, M, S, T, , event A, E, I; 4. , event has outcomes — Jan, Mar, May, Jul, Aug, Oct, Dec.)

Activity 2 — Sample Space Circuit (14 min)

Pairs. Every answer lists the sample space systematically and states .

Set A — list the sample space.

1. Rolling a twelve-sided die numbered –.
2. Drawing a counter from a bag containing red, blue and green counter, recording the colour.
3. The same bag, recording which counter (imagine them numbered).
4. Choosing a day of the week at random.
5. Drawing one card from a pack and recording its value ().

Set B — list the event’s outcomes and count them.

6. Die –; event: “a prime number”.
7. Day of the week; event: “a weekend day”.
8. Card value; event: “a picture card”.
9. Die –; event: “not a “.
10. Letter from PROBABILITY; event: “a letter appearing more than once in the word”.

Set C — the tricky ones.

11. Tossing a coin; event: “heads or tails”.
12. Rolling a die; event: “a number greater than “.
13. Bag of red counters only; sample space when recording colour.

Socratic scaffolding for Set C:

Prompt	Purpose
Q11: how many outcomes does this event contain?	Both — the event is the whole sample space. It is certain.
Q12: how many outcomes?	None. The event is impossible — an empty set.
Q13: how many outcomes in the sample space?	One: red. A chance experiment with a certain result.
So can an event have zero outcomes?	Yes — that is what “impossible” means mathematically.
Looking back	Certain and impossible are the two extremes; Lesson 122 gives them numbers.

(Answers: 1. , ; 2. red, blue, green, ; 3. six distinct counters, ; 4. ; 5. ; 6. , outcomes; 7. ; 8. J, Q, K, ; 9. , ; 10. B, I, ; 11. all — certain; 12. — impossible; 13. red, .)

Q2 versus Q3 deserves the board. The same bag gives a sample space of or of depending on what is recorded. This matters enormously in Lesson 122, because the six counters are equally likely but the three colours are not.

Activity 3 — Inquiry: Equally Likely or Not? (9 min)

Pairs, with equipment.

For each experiment, list the sample space and decide whether the outcomes are equally likely. Justify.

1. Rolling a fair die: .
2. Drawing from a bag of red and blue, recording colour: red, blue.
3. Tossing two coins and recording the number of heads: .
4. Spinning a spinner with one large sector and three small ones.

Socratic scaffolding:

Prompt	Purpose
Q1: is any face favoured?	No, if the die is fair — equally likely ✓
Q2: two outcomes. Equally likely?	No — three of the four counters are red. A two-outcome sample space does not mean a fifty-fifty chance.
Q3: list the ways each can happen.	heads: TT. head: HT or TH. heads: HH.
So is head as likely as ?	No — twice as likely, because two of the four equally-likely coin results give it.
Q4: what makes the spinner unfair?	Unequal sector sizes — area determines likelihood.
Looking back	Listing a sample space tells you what can happen, not how likely each is. Those are separate questions.

The misconception this kills, named on the board: “Two outcomes means a chance” is false. Whether outcomes are equally likely is a separate question from how many there are.

Checks for Understanding

(5 minutes — exit ticket)

1. Define “sample space” in one sentence.
2. List the sample space for drawing a letter from STATISTICS. State .
3. For rolling a die, list the outcomes in the event “an odd number greater than “.
4. Give an example of an impossible event when rolling a standard die.
5. Reasoning. A bag holds green and yellow ball. A student says “there are two outcomes, so each has a chance.” Explain the error.

Answers: 1. The set of all possible outcomes of a chance experiment; 2. A, C, I, S, T, ; 3. ; 4. E.g. “rolling a ” or “rolling a negative number”; 5. The two outcomes are not equally likely — five of the six balls are green, so green is far more probable. The number of outcomes says nothing about their likelihoods.

Common Misconceptions

Misconception	How to pre-empt it
Two outcomes means fifty-fifty.	Inquiry Q2 and exit Q5, named explicitly on the board.
Listing sample spaces randomly, losing outcomes.	Systematic listing (alphabetical or numerical) is required.
Repeating outcomes (PROBABILITY’s two B’s).	The no-repeats requirement, demonstrated.
Confusing an outcome with an event.	The definition table; events may contain several outcomes.
Believing an event must contain at least one outcome.	Set C Q12 — impossible events are empty.
Thinking the sample space is fixed by the equipment.	Warmup Q4 vs Q5; circuit Q2 vs Q3.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). How many outcomes are in the sample space when drawing one card from a standard pack and recording the exact card?

Answer

— thirteen values in each of four suits.

E2 (AMC Junior style). A spinner has sectors numbered to . List the outcomes in the event “a number that is both even and greater than “.

Answer

— three outcomes.

E3 (Challenge). A bag contains counters numbered to . How many outcomes are in the event “a multiple of or a multiple of “?

Answer

Multiples of : (six). Multiples of : (four). But is in both, so the event has outcomes.

E4 (Challenge). Two experiments use the same bag of red and blue counters. One records colour, the other records which counter. Explain why the sample spaces differ and which has equally likely outcomes.

Answer

Recording colour gives red, blue (, not equally likely); recording the counter gives six distinct outcomes (, equally likely if the draw is fair). The sample space depends on what is recorded, and only the finer one has equal likelihoods.

E5 (Challenge). For rolling a standard die, find an event containing exactly outcomes, and one containing exactly .

Answer

Four outcomes: e.g. “not a or a ” . Zero outcomes: e.g. “a number greater than “.

Homework

1. Define chance experiment, outcome, sample space and event, with a die example for each.
2. List the sample space and state : (a) rolling an eight-sided die (b) drawing a letter from GEOMETRY (c) choosing a vowel from the alphabet (d) spinning a spinner with sectors red, blue, green, yellow, purple.
3. For rolling a die –, list the outcomes in each event: (a) a factor of (b) a square number (c) not a multiple of (d) a number less than .
4. A bag has white, black and red marble. (a) List the sample space when recording colour. (b) State when recording which marble. (c) Are the colour outcomes equally likely? Explain.
5. For drawing one card from a pack: (a) list the sample space for the suit (b) how many outcomes in “a red picture card”?
6. Give an example, for a spinner numbered –, of: (a) a certain event (b) an impossible event (c) an event with exactly outcomes.
7. Explain why “win, lose” being the sample space for a raffle does not mean a chance of winning.
8. Reasoning. Explain why listing a sample space systematically matters.
9. Reasoning. Explain how the same physical experiment can have two different sample spaces.
10. Challenge. A bag holds counters numbered to . How many outcomes are in the event “a multiple of or a multiple of “?

Answers: Q2 — (a) , (b) E, G, M, O, R, T, Y, (c) A, E, I, O, U, (d) . Q3 — (a) (b) (c) (d) — impossible. Q4 — (a) white, black, red (b) (c) no — four of the eight marbles are white but only one is red. Q5 — (a) four suits (b) : J, Q, K in hearts and diamonds. Q7 — the two outcomes are not equally likely; with tickets and one prize, winning is far less likely than losing. Q8 — it guarantees completeness and prevents repeats; random listing reliably loses outcomes. Q9 — the sample space depends on what is recorded, not on the equipment: a die roll recorded as its number gives six outcomes, recorded as odd/even gives two. Q10 — multiples of : (seven); multiples of : (five); both: (two); total .