Single-outcome events, and why all of them together come to 1

What to assume they know

• Favourable over total: the definition this chapter runs on — the definition, and that an event is a collection of outcomes
• Adding fractions that share a denominator
• That a whole can be cut into parts in more than one way without changing the whole
• Recognising when a list of groups covers everything and when two groups share a member

What they should be able to do

• Define an elementary event and decide, for any event in the chapter, whether it is one
• Explain why an experiment with n outcomes has exactly n elementary events
• Total up what each elementary event carries, right across one experiment, and account for the answer
• Explain the total by pointing at the numerators rather than by checking the arithmetic
• State the two conditions a group of events must satisfy before its probabilities may be expected to total 1
• Show that the three marble colours of the chapter's Example 8 satisfy those conditions while none of them is elementary
• Produce a group of events whose probabilities do not total 1, and say which condition fails
• Complete the total for the eleven possible sums of two dice and check it against the same principle

Where it usually goes wrong

• "Every event is elementary." The chapter kills this on p. 206 by naming two events of Example 3 that are not, and giving their sizes. Any event described by a condition rather than by a single result is a candidate for holding several outcomes.
• "If two probabilities total 1 the events are elementary." Example 3's pair totals 1 and neither event is elementary. Totalling 1 is about covering the outcomes, not about being small.
• "Any list of events has probabilities totalling 1." The red card and the king total 30/52. The conditions are what earn the total, and the chapter states the result without stating them.
• "Nine marbles in three colours means three outcomes." There are nine outcomes. The three colours are three events of sizes 2, 3 and 4. Students collapse the marbles into their colours and then divide by 3.
• "The eleven sums of two dice are eleven equal chances." That is the whole point of question 22(ii), and the totals above show why it fails — the eleven events are a genuine partition but they are not the same size.
• "The total being 1 is a coincidence you check case by case." It is forced by the arithmetic: the numerators are the sizes of the parts and they add to the size of the whole.

Questions to check understanding

• Fill in what every elementary event's probability totals to, as in Exercise 14.1 question 1(iv)
• Decide whether a named event is elementary, and give its size in outcomes
• Complete a partial probability table and verify that it totals 1, as in question 22(i)
• Find a missing probability from the fact that a group totals 1
• Explain why a supplied group of events does not total 1 — the reasoning form the board uses to test whether the conditions were understood or the result was memorised

Examples worth working on the board

Values marked verified are worked out here on the chapter's printed data.

• The definition (p. 205, first remark). An event holding exactly one outcome. The page immediately classifies its own earlier work: both events of Example 1 and all three of Example 2 qualify.
• Example 1 revisited (p. 205). Head 1/2 and tail 1/2. Verified: the total is 1.
• Example 2 revisited (p. 205). Yellow, red and blue at 1/3 each. Verified: the total is 1. Two experiments, two different values of n, one total.
• The general claim (p. 205, second remark). Total up what every elementary event carries, across one whole experiment, and the answer is 1; the page says this holds generally rather than only in the two cases shown.
• The counter-case the chapter supplies itself (p. 206). In Example 3 the event of a die result above 4 holds two outcomes and the event of a result of 4 or below holds four, so neither is elementary — the page says so explicitly. This matters: those two probabilities also total 1, and if the explanation does not handle it the student concludes that totalling 1 is what makes an event elementary.
• Example 8 (p. 209). A box of 3 blue, 2 white and 4 red marbles, one drawn at random. Verified: nine outcomes; P(white) = 2/9, P(blue) = 3/9 = 1/3, P(red) = 4/9, and 2/9 + 3/9 + 4/9 = 1, which the page notes. None of the three events is elementary — they hold 2, 3 and 4 outcomes. This is the chapter's own demonstration that the total survives regrouping, and it is the centrepiece of sections 6 and 7.
• A group that fails, built from the chapter's own deck (p. 207 for the deck's structure). Take a red card and a king. Verified: 26/52 + 4/52 = 30/52, which is not 1 — the two events miss most of the pack and they overlap on the two red kings. Note that both conditions fail together here, so this pair on its own cannot show which failure produces which symptom. Two events that between them cover every card but overlap (a red card and a card that is not a black king, say), and two events with nothing in common that leave cards uncovered (a heart and a spade). Three groups, one clean lesson each. All three pairings are added here; the chapter never puts any of these events together.
• Exercise 14.1 question 22(i) (p. 216). A table of the eleven possible sums of two dice, 2 to 12, with three cells pre-filled: 1/36 at a sum of 2, 5/36 at a sum of 8, and 1/36 at a sum of 12. Verified by counting the cells of Fig. 14.3: the eleven counts run 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, giving 1/36, 1/18, 1/12, 1/9, 5/36, 1/6, 5/36, 1/9, 1/12, 1/18, 1/36, and these total 36/36. Eleven events of eleven quite different sizes — the two extremes hold a single ordered pair each, (1, 1) and (6, 6), and so are elementary events by the p. 205 definition, while the middle one holds six — and the total is still 1. The unequal sizes are the whole point: this is why the eleven cannot each be handed 1/11, which is exactly what part (ii) of the question asks the student to see.
• Exercise 14.1 question 1(iv) (p. 214) asks for this total as a fill-in-the-blank. Verified: 1.

Figures to have open

• Nine marbles drawn individually, then regrouped into bins of 3, 2 and 4. The whole argument of section 7 is that the outcomes do not change when the grouping does, and it cannot be made without seeing both states. The chapter prints no such figure. Must be built.
• A covering strip showing the two failure modes — a gap and an overlap — against a correct covering. Must be built; it is the picture that carries the two conditions the book never states.
• The 36-cell grid of Fig. 14.3 (p. 213) with the eleven anti-diagonals distinguishable, so the sums can be shaded one at a time. Redraw rather than reproduce; the chapter's own version already carries a ring drawn around the five cells that sum to 8.
• No photograph is needed.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 14 "Probability", §14.1, p. 205 — the two remarks that define an elementary event and state the total
• p. 206, the paragraph classifying the events of Example 3 as not elementary
• Example 8 and its closing observation, p. 209
• Fig. 14.3 and Example 13, pp. 212–213, used here only for the sums
• Exercise 14.1 questions 1(iv) and 22, pp. 214 and 216
• §14.2 Summary, p. 217, point 5
• The deck's structure, p. 207, used for the non-partition illustration