Listing every outcome: the sample space

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What to assume they know

Experimental probability: relative frequency over many trials — outcome and sample space as §7.2.1 first introduces them
Theoretical probability: counting favourable outcomes when all are equally likely — the equally likely assumption, and why counting depends on it
Writing a set inside braces with commas between its members, and counting its members
Multiplying two small numbers, for the size of a combined experiment
The words even, odd, integer and prime as used in the exercises

What they should be able to do

Write the sample space of a described experiment inside braces, and state its size n(S)
State the two rules the chapter gives for a sample space, say what each one prevents, and give a list that obeys both and is still not usable
Use the symbols S and n(S) correctly, and name an element of a sample space
Give two different sample spaces for the same situation and say which question each one suits
Explain why the grain of a sample space decides whether its elements are equally likely
Construct the sample space for a two-part experiment, such as a die with a coin, and state its size
Explain why a legal sample space can still be the wrong one to count with, using a bag of coloured balls
Identify a proposed list that fails to be a sample space, and say which requirement it breaks
Compute the size of a sample space from the experiment rather than by listing it out

Where it usually goes wrong

"The sample space is a property of the experiment." It is a property of the question you are asking about the experiment. The Think and Reflect on p. 167 exists to establish this.
"A bigger sample space is a better one." {No Rain, Drizzle, Light Rain, Heavy Rain} is better only if you care about how hard it rains.
"Every sample space has equally likely elements." {Win, Lose, Draw} does not. {0, 1, 2, 3} heads does not. Neither of those is a defective sample space.
"HT and TH are the same outcome." The chapter's own table separates them, and n(S) = 4 depends on the separation. Students who fuse them get 1/3 where the answer is 1/4.
"n(S) is the denominator." It is the denominator only when the elements are equally likely. This single sentence prevents most of the errors in the chapter's starred questions.
"An impossible outcome can be left in the list; it just gets probability 0." Not under this chapter's counting formula, where it inflates the denominator and makes every answer wrong.
"You have to write the list out to know its size." Six faces with two coin faces gives twelve without writing anything; four balls drawn twice with replacement gives sixteen.
"A ball's colour is an outcome." It can be, and if you choose it as one you have given up the right to count. Choose the balls.

Questions to check understanding

Write the sample space for a described experiment and state n(S)
State the number of outcomes in a sample space without listing them, from the structure of the experiment
Given a proposed list, say whether it is a sample space and which requirement it fails — the form End-of-Chapter Q15 takes, where the "why" carries the marks
Write two different sample spaces for one situation and say which question each suits
Fill in the blank naming the set of all possible outcomes (End-of-Chapter Q1 (ii))
Write the sample space for a combined experiment such as a die with a coin, or a coin with numbered cards
Reasoning question: why is the sample space for tossing two coins of size four and not three?

Examples worth working on the board

Inputs, not answers, except where the chapter prints the result itself. Values marked Verified are worked out here or an added reading of a printed page; the chapter prints no answers and this volume has no appended answer key.

The definition and its notation (p. 166). The chapter gives the letter S to the full list of outcomes; a single outcome sitting in it is called an element; and how many elements there are is the sample size, written n(S).
The three printed bullets (p. 166). The first requires that nothing possible be missing from S. The second forbids writing any one outcome twice over. The third is no rule at all — it is where n(S) gets defined. Read on the printed page: exactly three bullets, of which two are rules and one is a definition.
The five examples the chapter gives (p. 166).
- whether it rains tomorrow: S = {Rain, No Rain}, n(S) = 2
- a team's result in a match: S = {Win, Lose, Draw}, n(S) = 3
- tossing a fair coin once: S = {Heads (H), Tails (T)}, size 2
- rolling a standard 6-sided die: S = {1, 2, 3, 4, 5, 6}, size 6
- tossing two coins simultaneously: S = {HH, HT, TH, TT}, size 4
The two-coin table (p. 166). Beside the fifth example the page prints a three-column table headed Coin 1, Coin 2 and Outcome, with four rows: H and H giving HH, H and T giving HT, T and H giving TH, T and T giving TT. Read on the printed page. That table is the whole argument of section 6: HT and TH are different outcomes because the two coins are different coins, and n(S) = 4 rather than 3 depends on keeping them apart.
The Think and Reflect (p. 167). With S = {Rain, No Rain} you have committed to caring only whether it rains. To tell a drizzle apart from a downpour you have to widen the list into the four-item {No Rain, Drizzle, Light Rain, Heavy Rain}, and the chapter's moral is that the sample space has to be detailed enough for the problem being studied. Verified counts: n(S) = 2 and n(S) = 4 for the same tomorrow.
The observation the chapter stops just short of, and which section 8 supplies. Neither of those two rain sample spaces has equally likely elements, and nor does {Win, Lose, Draw}. So the chapter's own first two examples are sample spaces that you cannot apply the theoretical formula to. It never says this. It is the most important unstated fact in §7.3 and the reason this topic is worth thirteen minutes.
Exercise Set 7.3 Q1 (p. 167). How many outcomes are in the sample space of one roll of a 6-sided die? Verified: 6.
Exercise Set 7.3 Q2 (pp. 167–168). Write the sample space for each.
- (i) one die rolled together with one coin tossed. Verified: twelve elements — each of the six faces paired with each of the two faces of the coin, for instance written (1, H) up to (6, T). n(S) = 12, and 12 = 6 × 2.
- (ii) choosing a random integer between −5 and +5. Verified: the wording does not settle whether the two ends are included. Excluding them gives the nine integers from −4 to 4; including them gives eleven, from −5 to 5. Both readings are defensible.
- (iii) a box holding 5 green and 7 red balls, one ball drawn at random. Verified: two sensible sample spaces — {green, red} with n(S) = 2, or the twelve individual balls with n(S) = 12. The two colours are not equally likely: they come out 5/12 and 7/12. So the two-element list is a legitimate sample space that must not be counted with, which is section 8's case handed to you by the chapter's own exercise.
Exercise Set 7.3 Q3 (p. 168). At a village fair there are three snacks — samosa, pakora and bhaji — and two drinks, chai or lassi. List every snack-and-drink pair, then list the event of choosing samosa as the snack. Verified: 3 × 2 = 6 pairs; the samosa event holds 2 of them. The event half belongs to An event is a selection from the sample space; the six-element sample space is this topic's, and it is the cleanest instance of a size read off the experiment.
End-of-Chapter Q1 (ii) (p. 170). A fill-in-the-blank asking for the name of the set of all possible outcomes of a random experiment. Verified answer: sample space.
End-of-Chapter Q13 (iii) (p. 172). A box of 4 balls numbered 1 to 4; a ball is drawn and recorded, then either replaced or not, then a second is drawn. Asked for the sizes of the two sample spaces. Verified: 4 × 4 = 16 when the first ball goes back; 4 × 3 = 12 when it does not. The tree diagrams belong to Tree diagrams make the sample space of a two-step experiment visible; the two sizes belong here.
End-of-Chapter Q14 (p. 172). List the sample space for tossing a coin and drawing one of 6 cards numbered 1 to 6 at the same time. Verified: 12 elements, the same structure as Q2 (i), which makes it worth putting the two side by side.
End-of-Chapter Q15 (pp. 172–173) — section 10's whole reason for existing. Toss three coins; what gets written down is the count of heads, nothing else. Which of the four lists below is the sample space, and where do the rest fail? (i) {1, 2, 3}; (ii) {0, 1, 2}; (iii) {0, 1, 2, 3, 4}; (iv) {0, 1, 2, 3}. Verified: (iv) is right. (i) leaves out 0, so it breaks the first printed rule. (ii) leaves out 3, likewise. (iii) breaks neither printed rule — it contains every possible outcome and lists nothing twice — and yet it is not a sample space, because 4 heads cannot occur with three coins. So the chapter's two bullets, taken as written, do not exclude (iii). Every listed outcome must be one the experiment can actually produce. Read on the page images of pp. 166 and 172–173.
Why the missing rule matters arithmetically, not just tidily. Verified: with the list {0, 1, 2, 3, 4} the chapter's theoretical formula would give the probability of exactly two heads as 1 out of 5. The right answer, from the eight equally likely three-coin outcomes, is 3 out of 8. An impossible entry corrupts the denominator, so this is not a matter of neatness.
And the sharpest illustration of section 8, from the same question. Verified: the four elements of {0, 1, 2, 3} are not equally likely — their probabilities are 1/8, 3/8, 3/8 and 1/8. So End-of-Chapter Q15's correct sample space is one you cannot count with either, while End-of-Chapter Q4 (v) and Q12 (iii), which ask for probabilities about three coins, silently work in the eight-element space instead. Same experiment, two sample spaces, one countable and one not. Show the two lists together; it is the best thing in the chapter for this idea.

Figures to have open

The Coin 1 / Coin 2 / Outcome table (p. 166) redrawn and fillable row by row. This is the only printed figure §7.3.1 has, it is a plain table, and it carries section 6 entirely.
A one-tomorrow, two-lists panel for the Think and Reflect. Not in the book; the chapter prints the box as text.
A twelve-ball panel that can be shown as twelve objects and then collapsed to two colour groups, with 5/12 and 7/12 appearing on the groups. Not in the book, and the topic's most important addition.
A four-list comparison board for End-of-Chapter Q15, with the two printed rules shown as tick-boxes beside each list so that (iii) visibly passes both and still fails. Not in the book.
A grid of the eight three-coin outcomes beside the four-element head-count list, with the 1/8, 3/8, 3/8, 1/8 weights drawn. Not in the book.
No photograph is needed.

Where this sits in the book

NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 7, §7.3 "Elements of Probability: Sample Spaces and Events" and §7.3.1 "Sample Space" (pp. 166–167), comprising the definition and three bullets and the five examples on p. 166, the Coin 1 / Coin 2 / Outcome table on p. 166, and the Think and Reflect on p. 167.
The term is first introduced eight pages earlier, in §7.2.1 (p. 160), handled in Experimental probability: relative frequency over many trials; §7.3.1 opens by recalling it.
Exercise Set 7.3, Q1, Q2 and Q3 (pp. 167–168) — Q3's event half belongs to An event is a selection from the sample space.
End-of-Chapter Q1 (ii) (p. 170), Q13 (iii) (p. 172), Q14 (p. 172) and the starred Q15 (pp. 172–173).
The Chapter Summary (p. 173) restates the sample space as a set written with numbered outcomes, and prints neither rule and neither n(S) nor the sample size. Checked on the printed page.