Equally likely outcomes, and the everyday cases where that fails

What to assume they know

• Probability from Class IX as a fraction obtained by running an experiment many times and tallying how often something happened
• Reading a fraction as a part of a whole, and simplifying it
• The idea that a physical object can be symmetric — that a shape can have two sides with nothing to tell them apart
• Listing every possibility in a small situation without missing one or writing one twice

What they should be able to do

• State what is being assumed when a coin is called fair, and separate that from what is being assumed when the toss is called random
• Explain the symmetry argument that makes the two faces of a coin interchangeable, and say why it is an argument about the object rather than about the thrower
• List the outcomes of throwing one die and say why the chapter treats the six as interchangeable
• Show, using the chapter's bag of four red and one blue ball, that colours can fail to be equally likely while the individual balls do not
• Decide, for a described situation, whether the two named results are equally likely, and give the reason rather than the verdict
• Identify the standing assumption the chapter adopts and say what would be lost without it
• Explain why tossing a coin is accepted as a fair way to start a match
• Diagnose the error in an argument that assigns equal probabilities merely because two or three results have been named

Where it usually goes wrong

• "Two possible results, so each has probability one-half." This is the error the chapter closes on in question 25, and it is the reason this topic comes first. Two named results are two descriptions; they are equally likely only if they cover equal numbers of outcomes.
• "Fair means the person tossing is honest." Fairness here is a property of the object — its symmetry. Honesty of the throw is the second, separate condition the chapter calls a random toss.
• "Equally likely is a property of the experiment." It is a property of the outcome list. The chapter's bag is one experiment carrying an outcome list that is even and another that is not.
• "If the outcomes are not equally likely, nothing can be computed." The bag still yields 4/5 and 1/5 — by going back to the five balls, which are even. The fix for an uneven carving is a finer carving, not surrender.
• "A coin could land on its edge, so the model is wrong." The chapter dismisses this deliberately and names the surface on which it would not be dismissible. Choosing what to ignore is part of modelling, and doing it openly is what the page demonstrates.
• "The chapter proved that dice are fair." It did not; it declared it. Say so, because the difference between an assumption and a result is the whole subject of the next topic.

Questions to check understanding

• Given a described experiment, decide whether the named results are equally likely and justify the decision — the form of Exercise 14.1 question 2, and the form the board favours because it cannot be answered by computing
• Explain why a stated procedure counts as fair, as in question 3
• Judge a supplied argument as correct or not and give reasons, as in question 25
• Rewrite an uneven outcome list as an even one by going to a finer grain
• Distinguish, in one sentence, an assumption the chapter makes from a result it establishes

Examples worth working on the board

Values marked verified are worked out here on data printed inside pp. 202–203 and pp. 214–217. The chapter prints no answers.

• The coin (p. 202). Two results, head up or tail up. The page rules out a landing on the rim by argument, remarking that sand would be a setting where it could happen. The coin must be symmetric, and the toss must be free of interference. They are stated in one tinted box but they are two different assumptions.
• The die (p. 202). Results 1, 2, 3, 4, 5, 6. The page declares that a die in this book always means a fair one, so equal likelihood here is a convention the chapter adopts, not a conclusion it reaches.
• The bag (p. 203). Four red balls and one blue ball, drawn without looking. The page argues that red and blue are not equally likely and that drawing a ball — of whatever colour — is. Verified: on the five balls as outcomes, P(red) = 4/5 and P(blue) = 1/5. The chapter states the inequality but does not compute these two numbers; they are added here, and they make the point sharper than the verbal comparison does.
• Exercise 14.1 question 2 (p. 214), four situations to be judged with reasons: a driver tries to start a car, which starts or does not; a player shoots at a basket and scores or misses; a true-false question is answered rightly or wrongly; a baby is born a boy or a girl. The question asks for an explanation. An added reading: the car depends on the state of the car and is not even; the shot depends on the player's skill and is not even; the true-false answer is even only if it is a blind guess, which is what a trial with no knowledge means; a birth is treated as even to a good approximation. Say that the third and fourth are the arguable ones, and that saying why is the whole answer.
• Exercise 14.1 question 3 (p. 214): why a coin toss is accepted as fair for deciding which side starts a football match. The answer is the section-2 argument reused — the two results are interchangeable, so neither team is favoured before the game begins.
• Exercise 14.1 question 25 (p. 217), two arguments to be judged. (i) Two coins tossed together, described as ending both heads, both tails, or mixed, each supposedly 1/3. (ii) One die, described as landing odd or landing even, so odd supposedly 1/2. Verified: (i) is wrong — the four results are HH, HT, TH, TT, so the three descriptions carry 1/4, 1/4 and 1/2. (ii) reaches 1/2, which is correct, because the odd faces and the even faces number three each — but the reasoning offered is the same invalid reasoning as in (i) and only lands correctly by luck of the counts. This pair is the best single item in the chapter for this topic, precisely because one of them is right for the wrong reason.
• No figure is printed on pp. 202–203. Both pages were opened as page images to check this; the only non-text element on p. 202 is the chapter-opening QR code and the tinted box holding the definition of a fair coin.

Figures to have open

• The five balls of the bag drawn as five distinct objects, then regrouped into two colour bins of size four and one. This is the argument of sections 6 and 7 and the explanation cannot make it in words alone. The chapter prints no such figure — it must be built, and it is the single most important new picture in this brief.
• Six equal slots for the die faces, reusable in later topics as the denominator strip. Standard schematic.
• A two-panel comparison for question 25(i): four coin outcomes above, three verbal descriptions below, with two of the four arrows converging on "one of each". Standard schematic; must be built.
• No photograph is needed. The car, the basketball, the true-false question and the birth can each be a labelled icon carrying its pair of results.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 14 "Probability", §14.1, pp. 202–203 — the fair coin and its tinted box, the die, and the bag of five balls
• Exercise 14.1 questions 2, 3 and 25, pp. 214 and 217
• §14.2 Summary, p. 217, whose first point carries the equal-likelihood assumption as a clause of the definition itself
• Forward pointer inside the same chapter: the bag argument is re-used at Coins, dice, bags and a deck of 52: getting the denominator right, where the denominator is fixed by the same reasoning