When outcomes cannot be counted, measuring length or area instead

What to assume they know

• Favourable over total: the definition this chapter runs on — the definition as a ratio, and the equal-likelihood assumption it carries
• The impossible and the certain pin the scale at 0 and at 1 — that a probability lies between 0 and 1
• Reading a position on a number line, including a half
• Area of a rectangle, and area of a circle from its radius
• That a diameter is twice a radius
• Subtracting to recover an unlabelled length from a total and a part

What they should be able to do

• Explain why the counting definition cannot be applied as it stands when the outcomes fill an interval or a region
• Identify what the ratio was measuring, so that the replacement can be justified rather than announced
• Compute a probability as a ratio of lengths on a number line
• Compute a probability as a ratio of areas within a plane region
• Recover an unlabelled dimension of a figure by subtracting a labelled part from a labelled whole
• Argue that the position of the favourable region inside the whole does not affect the answer
• Recognise the printed marking that puts this material outside the examination, and describe its status honestly
• Restate equal likelihood in the language of an evenly spread region

Where it usually goes wrong

• "The lake is 3 km by 2 km." The 2 km arrow measures the strip below the lake, not the lake. This is the error the figure is practically designed to produce, and a reader who makes it cannot reconcile their own answer with the printed one. Teach the subtraction as a step, not as a detail.
• "Infinitely many outcomes means every probability must be 0." The share is what survives, and a share of 1/4 is perfectly ordinary. Nothing was divided by infinity.
• "Moving the lake changes the answer." It does not, and saying so is the best evidence that the method is measuring size rather than location.
• "Probability 0 always means the thing cannot happen." True while outcomes are counted, and it stops being safe here: the music stopping at exactly one minute occupies no length at all, yet the instant is a possible one. The chapter does not raise this. The impossible and the certain pin the scale at 0 and at 1 hands the point across.
• "The die has to fit inside the circle." The question treats the die as landing at a point. It is an idealisation and the chapter leaves it unstated; say so, because a student who pictures a real cube will start subtracting a margin.
• "Starred means unimportant." Starred means outside the examination. These three problems carry the only genuine extension of the definition in the whole chapter, and an explanation that skips them teaches the definition as narrower than it is.

Questions to check understanding

• Compute a probability as a ratio of two lengths on a marked interval
• Compute a probability as a ratio of two areas, with at least one dimension requiring recovery by subtraction
• Explain why the counting definition needs replacing before either computation can be made
• Leave an answer in terms of π where a circle is involved, as question 20 requires
• Note for anyone: because these three items are marked outside the examination, expect them in classwork and enrichment rather than in a board paper. Pitch the explanation accordingly and say so.

Examples worth working on the board

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

• The observation (p. 210). The chapter pauses to note that every experiment worked so far had finitely many outcomes, then raises two that do not: one where the result is any value lying between two given ones, and one where it is any point inside a circle or a rectangle. It says plainly that the definition as learnt cannot be used in that form. Keep the pause — it is the hinge of the topic and the reason the two examples exist.
• Example 10 and Fig. 14.1 (pp. 210–211). Music is to be stopped at some instant within 2 minutes of starting, and the question asks for the chance that it stops within the first half-minute. The figure is a plain number line running from 0 to 2 with ticks at 0, ½, 1 and 2 and an arrowhead at the right end. Verified: the whole interval measures 2, the favourable stretch measures ½, so the probability is (½)/2 = 1/4. The chapter's own bridge sentence after this asks whether the same idea extends from length to area.
• Example 11 and Fig. 14.2 (pp. 211–212). A helicopter has come down somewhere in a rectangular region, and the question asks for the chance that it is in the lake. The figure's labelling is the trap and must be handled explicitly. Four measurements are printed: 9 km along the bottom, 4.5 km up the left side, 6 km along the top, and 2 km up the right side. The 9 and the 4.5 give the whole rectangle. The 6 km arrow runs from the left edge only as far as the lake's near side, and the 2 km arrow runs from the lake's lower edge down to the base — so both of them measure what is outside the lake. The lake's own sides are nowhere printed. Verified by subtraction: the lake is 9 − 6 = 3 km across and 4.5 − 2 = 2.5 km deep. Verified: the region measures 4.5 × 9 = 40.5 square kilometres and the lake 2.5 × 3 = 7.5, so the probability is 7.5/40.5 = 75/405 = 5/27. This is the single most producer-critical fact in the brief: an explanation that reads 3 by 2 off the two labelled arrows will compute 6 square kilometres and be wrong, and the printed solution's 7.5 will then look like a misprint.
• Position against size. Verified: slide the same 3 by 2.5 lake anywhere inside the rectangle and the ratio is unchanged at 5/27, because neither area moves. This is the claim of section 9. It is added here — the chapter computes one arrangement and never remarks on it — and it is the cleanest way to show that area has taken over the role the count used to play.
• Exercise 14.1 question 20 and Fig. 14.6 (p. 216). A die is dropped at random onto a rectangular region measuring 3 m by 2 m, inside which a circle of diameter 1 m is drawn, and the question asks for the chance of landing inside the circle. Verified: the rectangle is 6 square metres; the radius is 0.5 m so the circle is π × 0.25, which is π/4 square metres; the probability is (π/4)/6 = π/24, about 0.131. The circle is drawn roughly central in the printed figure, which section 9 says does not matter.
• The printed status marker. Example 10, Example 11 and question 20 each carry a star, and each of the three pages sets a footnote below a rule at the foot of the page saying the item lies outside the examination. Confirmed on pp. 210, 211 and 216. Nothing else in Chapter 14 is marked this way.

Figures to have open

• A redrawn Fig. 14.2 (p. 211): a rectangle 9 wide by 4.5 tall with a lake in one corner, carrying the four printed arrows in their printed roles — the 6 km and the 2 km measuring outside the lake, not inside it — and then a second state in which the subtractions are performed and the lake's own 3 km and 2.5 km appear. The two-state version is essential. The chapter's own artwork is dense forest illustration; redraw it as a schematic and keep only the geometry.
• The same rectangle with the lake shown in three positions, for section 9. Must be built; it makes the topic's thesis visible in one frame.
• A redrawn Fig. 14.1 (p. 210): a number line from 0 to 2 with the ticks the book prints and the first half-unit shaded. Standard schematic.
• A redrawn Fig. 14.6 (p. 216): a 3 by 2 rectangle with a circle of diameter 1 inside it. Standard schematic.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 14 "Probability", §14.1, pp. 210–212 — the finiteness observation, Example 10 with Fig. 14.1, and Example 11 with Fig. 14.2
• Exercise 14.1 question 20 with Fig. 14.6, p. 216
• The footnotes marking all three as outside the examination, at the foot of pp. 210, 211 and 216
• Backward pointers: the definition being extended is Favourable over total: the definition this chapter runs on; the reading of a probability of 0 comes from The impossible and the certain pin the scale at 0 and at 1