Favourable over total: the definition this chapter runs on

What to assume they know

• Equally likely outcomes, and the everyday cases where that fails — equal likelihood as an assumption about the outcome list
• Why repeating an experiment is often impossible, and what replaces it — why a definition that avoids trials is worth having
• Listing every possibility in a small experiment, exhaustively and without duplication
• Simplifying a fraction to lowest terms, and recognising when two fractions are the same number
• That a whole made of equal parts is recovered by adding the parts

What they should be able to do

• State the definition in terms of what each of its two counts counts, and say which clause of it carries the assumption
• Justify the definition by assigning weight 1/n to each of n interchangeable outcomes and adding
• Compute the probability of a single-outcome event, as in the coin and the three balls
• Compute the probability of an event covering several outcomes, as in a die result above 4
• Translate an event described in words into an explicit list of the outcomes that favour it, before dividing anything
• Explain why the denominator never changes between parts of the same question while the numerator does
• Say what the definition does not require — no trials, no history, no repetition
• Place the definition historically, naming Laplace and the year the chapter attaches to it

Where it usually goes wrong

• "Probability is favourable over unfavourable." It is favourable over everything. In Example 3 the wrong reading would give 2/4 for a result above 4 instead of 2/6. Students who have met odds elsewhere import this constantly.
• "Two possible answers, so one-half." Example 3 is the chapter's own refutation: the event and its denial are each one description, and they come out 1/3 and 2/3. Counting descriptions is not counting outcomes.
• "The denominator is whichever number the question mentions." It is the size of the outcome list, fixed once for the whole experiment. Both parts of Example 3 divide by 6.
• "Balls of the same size is just scene-setting." It is the clause that makes the three outcomes interchangeable. Delete it and the computation is not justified.
• "You have to run the experiment to know the answer." Nothing in Example 1 was tossed. That is the point of the previous topic and this is where it pays.
• "Probability is a fraction, so it cannot be a decimal or a percentage." The chapter later prints 0.88 and 0.96 for the same kind of quantity. The ratio is a number; how it is written is a separate matter.

Questions to check understanding

• Compute the probability of a described event from an explicitly listed outcome set, showing the list before the fraction
• Both parts of a question sharing one experiment, so that a shifting denominator is exposed
• "Why is this answer what it is" questions, which the chapter models by printing its own unanswered prompts beside Example 1 and Example 4
• Identify the assumption a computation rests on, and say where the question states it
• One-mark recall of the definition, including its final clause — the clause is the part students drop

Examples worth working on the board

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

• The definition (p. 203). Two counts, one over the other: how many outcomes would make the event happen, and how many outcomes the experiment has altogether. The trailing clause granting equal likelihood is part of the definition, not a footnote to it.
• The justification the chapter does not print. With n interchangeable outcomes, no one of them can carry more weight than another, and the weights must fill the experiment, so each is 1/n. An event holding k of them therefore carries k × 1/n. This is the argument of section 3 and it is added here — the book asserts the ratio without decomposing it. It is worth the two minutes because every later result in the chapter is a consequence of it.
• Example 1 (p. 204). A coin tossed once. Two outcomes, head and tail; one favours a head. P(head) = 1/2, and P(tail) = 1/2 by the same count. The page asks the reader why the second answer follows and leaves the reason unwritten.
• Example 2 (pp. 204–205). A bag holding one red, one blue and one yellow ball, all the same size, one drawn without looking. Three outcomes; each of the three events is favoured by exactly one. P(yellow) = P(red) = P(blue) = 1/3. Note: the phrase about the balls being the same size is the guarantee of equal likelihood, not a decoration — it is why the draw cannot prefer one ball.
• Example 3 (p. 205). One die thrown once. Six outcomes. For a result above 4 the favourable outcomes are 5 and 6, so verified: 2/6 = 1/3. For a result of 4 or below they are 1, 2, 3 and 4, so verified: 4/6 = 2/3. Both parts share the denominator 6; only the numerator moves. This is the pair to show when section 9 lands.
• The history box (p. 204, a tinted block with a portrait). Probability theory traced to the sixteenth century and to J. Cardan, an Italian doctor who also worked in mathematics and who produced the subject's earliest book, about gambling games. James Bernoulli 1654–1705; A. de Moivre 1667–1754; Pierre Simon Laplace 1749–1827, whose 1812 work the chapter calls the greatest single contribution to the subject. The definition itself is attributed to Laplace in 1795 (p. 203). The block closes by listing fields where probability is now used: biology, economics, genetics, physics and sociology.
• The portrait (p. 204). An oval portrait of Laplace with his dates printed beneath it, set into the right-hand side of the tinted block. Confirmed on the printed page.

Figures to have open

• A weight strip: n equal cells each labelled 1/n, with k of them shaded and the shaded total shown alongside. This is the argument of section 3, it is the spine of the whole chapter, and the book prints nothing like it. Must be built.
• A two-column panel with outcomes on the left and descriptions on the right, so that the many-to-one arrows are visible. Must be built; reused from Equally likely outcomes, and the everyday cases where that fails.
• A six-cell die strip with two cells shaded, then four, over an unchanged denominator. Standard schematic.
• The Laplace portrait is printed in the chapter (p. 204). It is a period image rather than a diagram; the explanation can use a stylised drawing instead, and nothing in the mathematics depends on it.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 14 "Probability", §14.1, p. 203 — the definition and its attribution to Laplace in 1795
• The tinted history block and portrait, p. 204
• Examples 1, 2 and 3, pp. 204–205
• §14.2 Summary, p. 217, point 1
• Backward pointers: the assumption is Equally likely outcomes, and the everyday cases where that fails, the reason for wanting a trial-free definition is Why repeating an experiment is often impossible, and what replaces it