Why repeating an experiment is often impossible, and what replaces it

What to assume they know

• Equally likely outcomes, and the everyday cases where that fails — what it means to call outcomes equally likely, and that this is an assumption about the set-up
• The Class IX treatment of probability as a fraction obtained by tallying repeated trials
• Reading a fraction as a proportion, and comparing two fractions
• The idea that a fraction obtained from a small sample can move about, while one obtained from a large sample tends to settle

What they should be able to do

• Write down the Class IX formula in terms of what each of its two counts measures, and say why both counts require the experiment to have been performed
• Explain why the empirical route works comfortably for coins and dice
• State the limitation the chapter puts on repeating an experiment — that it may be too expensive or simply not feasible — and name the two cases it offers
• State what is bought and what is paid for when an assumption replaces repeated trials
• Distinguish the empirical and the theoretical probability of the same event by saying what each one is a statement about
• Explain why the two are expected to approach each other as trials pile up, and why the chapter stops short of promising it
• Identify, for a described situation, whether the empirical route or the theoretical route is available

Where it usually goes wrong

• "The theoretical value is the true one and the experimental value is a bad copy of it." The chapter does not say this. The computed value is only as true as its assumption. Take a coin that is secretly weighted: assuming it symmetric still yields a computed one-half for a head, and that one-half is worth exactly what the symmetry assumption is worth, which here is nothing. Where the assumption fails, the tallied value is the one telling the truth.
• "You can always just do the experiment more times." The earthquake is in the chapter precisely to kill this. Some experiments are not ours to run once, let alone often.
• "The two formulas are the same formula." They share a shape and share nothing else. One divides trials by trials, the other divides possibilities by possibilities. Students who see only the shape will happily put a count of trials over a count of outcomes.
• "Enough trials guarantee the two agree." The printed note says they may be expected to be nearly the same, and that hedge is doing real work. A run of trials can be long and still unlucky.
• "Theoretical probability means the answer is only a theory." It means the number was reasoned to rather than measured. Reasoning is not weaker here; it is what makes the satellite question answerable at all.
• "This section is background reading before the real chapter starts." It is the justification for everything that follows. Every later example computes without experimenting, and this is the page that says why that is allowed.

Questions to check understanding

• Given a described situation, say whether its probability could be found by repeated trials, and if not, why not
• State the difference between the experimental and the theoretical probability of one named event
• Explain why an assumption is required before the theoretical route can be taken
• Say what the closing note claims about long runs, and what it stops short of claiming
• Short reasoning questions of the "why is this method used here" kind, which is how the board examines a passage that carries no arithmetic

Examples worth working on the board

The chapter offers no numbers on this topic — it is the one stretch of Chapter 14 that argues rather than computes.

• The Class IX formula (p. 203). Numerator: how many of the trials ended in the event. Denominator: how many trials were run. Both are records. Point the explanation at the contrast with the definition arriving on the same page, whose two numbers are counts of possibilities rather than counts of trials — the two formulas look alike on the page and are about different things.
• Where repetition is fine (p. 203). The chapter names coin tossing and die throwing as the cases where the empirical route worked well. Use them as the baseline before the obstacles land.
• The satellite (p. 203). Computing how often a launch fails would mean launching repeatedly.
• The earthquake (p. 203). Computing how often a multi-storeyed building is destroyed would mean repeating the earthquake.
• How the page actually frames these two. Worth being exact, because the explanation will sound like it is quoting. The page raises only two limitations on repeating an experiment — the expense, and whether it is feasible at all — and then offers these two situations as ones where repetition is problematic; it does not assign the money objection to the satellite and the impossibility objection to the earthquake. Splitting them that way is an added reading, and a good one — take the satellite first as the case about cost and the earthquake second as the case about the world not being ours to re-run — but present it as an ordering the explanation is imposing.
• The closing note (p. 217, printed as a tinted block titled "A Note to the Reader"). It sets the two probabilities against each other — one is about what did happen, the other predicts from assumptions — and says that as trials increase the two may be expected to come close. The hedge is in the printed wording and must survive into the explanation.
• An illustration the explanation may build, clearly labelled as invented. Toss a fair coin in runs and show the tallied proportion wandering near 1/2 and settling as the runs lengthen, against the computed 1/2 drawn as a fixed line. No such data is printed in Chapter 14. If the explanation shows numbers here they are not in the book and must be captioned that way; do not let a simulated run be read as the textbook's.
• No figure is printed on p. 203. The page was opened as a page image to check this; it carries two displayed formulas and no artwork.

Figures to have open

• A long-run convergence picture: trials along the horizontal axis, the tallied proportion as a jagged line, the computed value as a straight one, the jagged line settling towards it without ever locking on. The chapter prints no such figure and the closing note cannot be taught without it. Must be built, and the caption must say the data is illustrative.
• A side-by-side of the two formulas with their numerators and denominators colour-coded by what they count. Standard schematic, built from p. 203.
• Icons only for the satellite and the earthquake; no photograph is needed, and a photograph would distract from the point, which is about repetition rather than about disasters.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 14 "Probability", §14.1, p. 203 — the Class IX formula, the limits of repetition, the satellite and the earthquake, and the arrival of the theoretical definition
• The chapter's closing tinted block "A Note to the Reader", p. 217, which owns the experimental-against-theoretical contrast for this topic
• §14.2 Summary, p. 217, point 1, which restates the theoretical definition with its assumption attached
• Backward pointer: the assumption itself is Equally likely outcomes, and the everyday cases where that fails; the definition that the assumption unlocks is Favourable over total: the definition this chapter runs on