The impossible and the certain pin the scale at 0 and at 1

What to assume they know

• Favourable over total: the definition this chapter runs on — the definition as a ratio of two counts
• Single-outcome events, and why all of them together come to 1 — that events are collections of outcomes and that a full partition totals 1
• That a fraction is 0 exactly when its numerator is, and 1 exactly when its two parts are equal
• Converting between a fraction, a decimal and a percentage
• Comparing signed numbers, so that a negative value can be recognised as outside a range

What they should be able to do

• Compute the probability of an event no outcome favours, and name what the chapter calls such an event
• Compute the probability of an event every outcome favours, and name it
• Derive the range 0 to 1 from the definition rather than quoting it
• Explain why the numerator can never exceed the denominator here
• Decide which of a set of candidate numbers could be a probability and justify each rejection
• Recognise a percentage as a probability written differently, and convert it
• Produce both ends of the scale from one described experiment
• Locate the two ends inside a larger experiment, such as the sums of two dice

Where it usually goes wrong

• "Probability 0 means the thing is barely possible, and 1 means it is very likely." In this chapter's finite setting 0 means no outcome favours the event at all, and 1 means every one does. They are counts hitting their limits, not strong opinions.
• "A percentage cannot be a probability." 15% is 0.15 and sits comfortably inside the range. The chapter itself prints probabilities as decimals — 0.88 and 0.96 in Example 12 — so the form of the number was never the test.
• "A very likely event could have probability 1.2." The numerator is a count drawn from the denominator's list. There is nothing for it to exceed the denominator with.
• "A negative probability just means the event works against you." Both counts are counts of things. Neither can be negative, so neither can their ratio.
• "0 and 1 are two extra facts to memorise." They are the endpoints of one argument. If the explanation teaches them as a list it has taught the wrong thing.
• "0 always means it cannot happen." True everywhere in this topic, and it stops being true in the same chapter: in Example 10 the favourable stretch for the music stopping at one exact instant has length 0, and the instant is still a possible one. The chapter does not raise this. Flag it as the edge of the claim and hand it to When outcomes cannot be counted, measuring length or area instead, rather than letting the explanation overstate the rule here.

Questions to check understanding

• Fill-in-the-blank recall of the two named events and their values, as in Exercise 14.1 question 1
• Given several candidate numbers, choose the one that cannot be a probability and say why — the multiple-choice form, and question 4 is its model
• Produce an impossible event and a certain event for a stated experiment
• Compute a probability that comes out to exactly 0 or exactly 1 from a real count
• Justify the range from the definition in a two-mark answer, which is the form that separates memorising from understanding

Examples worth working on the board

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

• The die asked for an 8 (p. 206). Six outcomes, none of them an 8, so the favourable count is zero. Verified: 0/6 = 0. The chapter draws the conclusion on the following page and attaches the name there.
• The die asked for a result below 7 (p. 207). Every face qualifies, so the favourable count equals the total. Verified: 6/6 = 1.
• The note that establishes the range (p. 207). The upper count can never beat the lower one, because the favourable outcomes are taken from the same list the lower count measures. The inequality follows immediately. Show the reason, not just the inequality — the reason is one sentence and it is what makes the bounds unforgettable.
• Example 13, parts (ii) and (iii) (p. 213). Two dice, 36 outcomes. A sum of 13 has no favourable cell at all, so verified: 0/36 = 0. A sum of at most 12 is favoured by every cell, so verified: 36/36 = 1. Same two ends, on an experiment thirty-six times larger — this is the pair that stops students believing 0 and 1 are properties of small experiments.
• Exercise 14.1 question 4 (p. 214). Four candidates: 2/3, −1.5, 15% and 0.7. Verified: 2/3 is about 0.667 and lies inside the range; 15% is 0.15 and lies inside it; 0.7 lies inside it; −1.5 is negative and no ratio of two counts can be, so it is the only one that cannot be a probability.
• Exercise 14.1 question 6 (p. 214). A bag holding only lemon-flavoured candies, one taken out without looking. Verified: an orange one has no favourable outcome, so 0; a lemon one is favoured by every outcome, so 1. One question, both ends, no arithmetic — the cleanest item in the chapter for this topic.
• Exercise 14.1 question 12, parts (i) and (iv) (p. 215). A spinner whose arrow settles on one of 1 to 8, the eight being equally likely, drawn as Fig. 14.5 — an eight-sector disc with the numbers set inside the sectors and an arrow drawn from the centre. Verified: an 8 gives 1/8; a result below 9 is favoured by all eight sectors, so 1.
• Exercise 14.1 question 1, parts (ii), (iii) and (v) (p. 214) ask for these values and names as fill-in-the-blanks. Verified: 0 with the name for an event that cannot happen; 1 with the name for one that must; and the two ends of the range.

Figures to have open

• A 0-to-1 number line as the explanation's standing scale, with the two ends labelled and room to drop candidate values onto it. The chapter prints the bounds only as an inequality; the line is an added device and it should recur whenever a probability is computed anywhere in this chapter.
• A growing-selection strip: the same outcome list with the shaded part running from empty to full, with the fraction updating beside it. This is the argument of section 6 and it must be built.
• The 36-cell grid from Fig. 14.3 (p. 213), shown fully unshaded and fully shaded. Redraw rather than reproduce.
• Fig. 14.5 (p. 215), the eight-sector spinner, is genuinely needed for question 12 and is a simple schematic to rebuild — eight equal sectors, numbers inside them, an arrow pivoted at the centre.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 14 "Probability", §14.1, pp. 206–207 — the two questions about a die, the two named events, and the note establishing the range
• Example 13 parts (ii) and (iii), p. 213
• Exercise 14.1 questions 1, 4, 6 and 12, pp. 214–215, with Fig. 14.5 on p. 215
• §14.2 Summary, p. 217, points 2, 3 and 4
• Forward pointer inside this chapter: the one place the reading of 0 needs care is When outcomes cannot be counted, measuring length or area instead, where outcomes stop being countable