An event is a selection from the sample space

What to assume they know

• Listing every outcome: the sample space — the sample space, its elements, and n(S)
• Theoretical probability: counting favourable outcomes when all are equally likely — the theoretical formula and the equally likely assumption
• The word subset, and the idea that a set can be selected from inside another
• Counting the members of a listed selection
• Reading conditions such as "greater than", "at least", "exactly" and "not"

What they should be able to do

• State what an event is, and write a described event as a selection from a named sample space
• Translate an "at least", an "exactly" and a "not" condition into three different subsets
• Compute the probability of an event by counting its elements against the sample space, when the elements are equally likely
• Identify the whole sample space and an empty selection as events, and give their probabilities
• Compute the probability of a "not" event by counting, without a complement rule
• State the fact that makes a complement rule work, and say that this chapter does not print it
• Give an event of probability 1 and an event of probability 0 for a described experiment
• Explain why an event's element count cannot always be divided by the sample size, using the chapter's fruit basket

Where it usually goes wrong

• "'At least one head' means exactly one head." The commonest error in the chapter. The wrong subset gives 1/2 and the right one gives 3/4.
• "An event is one outcome." It can be, and usually is not. The chapter's definition allows both and its first example uses three outcomes.
• "Any subset you can name has a probability you can get by counting." Only in a sample space whose elements are equally likely. The fruit basket is the chapter's own counter-instance.
• "The event is the answer." The event is a set. The probability is a number computed from it. Two separate deliverables, and exercises ask for each separately.
• "'Not red' needs a special rule." Count the balls that are not red. The subtraction is a shortcut, and in this chapter it is a shortcut that rests on something unstated.
• "Probability 0 means the event is not in the sample space." It means no element of the sample space satisfies it. The event is perfectly well described; it simply never happens.
• "An outcome can have probability 1." Only if the sample space has just one element. With two dice, every outcome sits at 1/36 — which is why End-of-Chapter Q11 has to be read as asking about events.
• "Adding a condition can only make an event more likely." Q12 (iii) shows the reverse: requiring the first coin as well as the total cuts 3/8 down to 2/8.

Questions to check understanding

• Write a described event as a subset of a named sample space
• Compute the probability of an event by counting, in an equally likely sample space
• Handle a "not" condition, both by direct counting and by subtracting from 1
• Give an event with probability 1 and an event with probability 0 for a stated experiment — the form End-of-Chapter Q11 intends
• Distinguish "at least", "exactly" and "at most" versions of the same question and give all three answers
• Compute a probability for an event defined by two conditions at once
• Reasoning question: a student writes the event "at least one head" as a two-element subset. Which outcome has been left out, and what does the answer become?

Examples worth working on the board

Inputs, not answers, except where the chapter prints the result itself. Values marked Verified are worked out here; the chapter prints no answers and this volume has no appended answer key.

• The definition (p. 167). The chapter allows an event to be one lone result, or several results taken together, from what a random action might produce; it glosses this as picking out particular outcomes from everything available, and then states the formal version — an event is a subset of a sample space. Note that a single lone result is admitted, so a one-element event is still an event, which students doubt.
• The chapter's three examples (p. 167).
  • Two coins. S = {HH, HT, TH, TT}. The event "at least one coin shows a head" is E = {HH, HT, TH}. Verified: three elements of four equally likely ones, so 3/4 = 0.75.
  • One 6-sided die. S = {1, 2, 3, 4, 5, 6}. The event "the number rolled is above 4" is E = {5, 6}. Verified: 2/6 = 1/3.
  • A fruit basket. S = {Apple, Banana, Orange}. The event "the fruit picked is yellow" is E = {Banana}.
• The fruit basket is the one to be careful with, and section 10 is about it. The chapter writes the subset and computes no probability at all — read on the printed page. And it is right not to: nothing says the basket holds one of each fruit. If it holds one apple, one banana and four oranges, the yellow event still has one element of the three-element sample space and its probability is 1/6, not 1/3. State the subset, note that the chapter stops there, and say what extra information would be needed. This is the cleanest demonstration in the chapter that writing the event and computing its probability are two different acts.
• The element people drop, for section 2. Students routinely read "at least one head" as "one head", giving {HT, TH} and 1/2. The correct subset includes HH, because two heads is one or more heads. Show the wrong subset first and let the count kill it — 2/4 against 3/4 — because the printed answer alone does not teach the error.
• The two extreme events, for section 4. Verified: the whole sample space is itself an event, satisfied by every outcome, with probability n(S)/n(S) = 1; a selection satisfied by no outcome has no elements and probability 0. End-of-Chapter Q1 (i) and (iii) ask for exactly these two values as fill-in-the-blanks (pp. 169–170), and End-of-Chapter Q9 (iv) (p. 171) supplies a concrete case of the first: on an eight-sector spinner numbered 1 to 8, the event "a number below 9" holds all eight sectors, so its probability is 8/8 = 1.
• End-of-Chapter Q4 (p. 170), read as five translations. Verified:
  • (i) two coins tossed together, at least one head — E = {HH, HT, TH}, so 3/4
  • (ii) ten identical cards numbered 1 to 10, one drawn, an even number — E = {2, 4, 6, 8, 10}, so 5/10 = 1/2
  • (iii) one roll of a die, a number above 4 — E = {5, 6}, so 2/6 = 1/3
  • (iv) a bag of 3 red, 2 blue and 1 green ball, one picked, not red — the six balls are the equally likely outcomes and three of them are not red, so 3/6 = 1/2
  • (v) three coins tossed at once, exactly two heads — in the eight-element sample space the qualifying outcomes are HHT, HTH and THH, so 3/8
• End-of-Chapter Q8 (p. 171), the PEACE cards. Five cards carrying P, E, A, C, E; one drawn without looking. Verified: (i) the event "P, E or C" holds P once, E twice and C once, so 4 of 5; (ii) "not an E" holds P, A and C, so 3 of 5. Note that (i) and (ii) between them cover a nice structural point: the two events overlap, and neither is the other's opposite.
• Section 8's missing rule, and why the chapter can do without it. Verified: for a "not" event you can either count what qualifies — three not-red balls out of six — or subtract: 1 − 3/6. The subtraction works because the probabilities of all the outcomes in a sample space total 1, which is a fact this chapter never states on any page from 155 to 173. So the chapter's route through Q4 (iv) and Q8 (ii) is direct counting, and it is legitimate.
• End-of-Chapter Q11 (p. 172), and it does not say what it means. Two dice of six faces each are thrown; the question wants an event whose probability is 0, and an outcome whose probability is 1. Verified: the sample space has 36 equally likely elements, so every single outcome has probability 1/36 and no outcome has probability 1. What has probability 1 is an event — the whole sample space, for instance "the sum is somewhere between 2 and 12". An event of probability 0 is easy: "the sum is 1", or "both dice show a 7". Since §7.3.1 defines an outcome as one element of S, the printed question asks for something that cannot exist.
• Exercise Set 7.3 Q3 (ii) (p. 168). At a village fair with three snacks and two drinks, list the event of choosing samosa as the snack. Verified: the sample space holds 6 snack-and-drink pairs and the samosa event holds 2 of them, so 2/6 = 1/3 if the six pairs are taken as equally likely. This is the chapter's only exercise that asks for an event to be listed rather than measured, which makes it the right drilling question for section 1.
• End-of-Chapter Q12 (iii) (p. 172), a two-condition event. Three coins tossed; the first must show a head, and the heads must total two. Verified: in the eight-element sample space the qualifying outcomes are HHT and HTH, so 2/8 = 1/4. Worth including because it is the chapter's only event defined by two conditions joined together, and it shows that adding a condition shrinks the subset.
• The notation, for section 11. The chapter writes events as E and defines n(S) for the sample space's size, and the Summary writes the probability of an event as P(E). It never writes a symbol for the size of an event.

Figures to have open

• A translation panel: a sentence in ordinary words on top, the braced subset beneath, and the sample space drawn as a row of cells with the chosen ones shaded. Reusable across sections 1, 2, 3 and 6, and it is the explanation's core visual. The chapter prints §7.3.2 as text only.
• The four two-coin outcomes as four cells, with the "at least one head" selection shaded and the tempting two-cell selection shown first and rejected. Not in the book.
• The eight three-coin outcomes as eight cells, so that "exactly two heads" and Q12 (iii)'s double condition can be shaded on the same board. Not in the book; this sample space is never printed in the chapter.
• A 6 × 6 grid of the 36 two-dice outcomes for section 9, so that "everything" and "nothing" can be ringed. Not in the book, and needed because End-of-Chapter Q11's flaw is only visible once the 36 cells are shown.
• The three-fruit basket drawn twice with different contents and the same subset. Not in the book, and the topic's most important addition.
• The five PEACE cards in a row (p. 171); redraw the printed row of boxed letters.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 7, §7.3.2 "Events" (p. 167), comprising the definition and the three worked examples of sample spaces with events.
• §7.3.1 (p. 166) supplies S, element and n(S), handled in Listing every outcome: the sample space.
• Exercise Set 7.3 Q3 (ii) (p. 168), whose sample-space half belongs to Listing every outcome: the sample space.
• End-of-Chapter Q1 (i) and (iii) (pp. 169–170), Q4 (p. 170), Q8 (p. 171), Q9 (iv) (p. 171), and the starred Q11 and Q12 (iii) (p. 172).
• The Chapter Summary (p. 173) restates the event as one outcome or a group of outcomes picked out of everything that could happen, and introduces the symbol P(E) in its second bullet. It prints no complement rule and no statement that the outcomes' probabilities total 1. Checked on the printed page.