The 0-to-1 scale, and what the endpoints mean

What to assume they know

• Probability as a measurement, not a guess — probability as a measurement, and the five ordinary-language labels
• Placing a rational number on the line, and distance as |a − b| — placing a number on the number line, and reading positions between two marks
• Converting between a fraction, a decimal and a percentage: 3/4, 0.75, 75%
• That a fraction cannot exceed 1 when its numerator is at most its denominator
• Counting a subset of a small collection

What they should be able to do

• State the range probability is measured on, and what a value of 0 and a value of 1 each mean
• Read a probability given as a decimal and restate it as a percentage, and the reverse
• Explain why no probability can be negative and none can exceed 1
• Read Fig. 7.1: state how many purple cards each of the five decks holds, and compute the probability for each
• Explain why the probability moves up the scale as the number of purple cards grows, with the deck size fixed
• Place the five printed labels — impossible, less likely, even chance, more likely, certain — in order on the scale, and say which of them name exact values
• Classify each of the chapter's five sample events against the scale and justify it from the counts given
• Use the Summary's notation P(E) and state the inequality it satisfies
• Rank four everyday events on the scale with reasons (Exercise Set 7.1)

Where it usually goes wrong

• "A probability can be 150%, or −0.2." It cannot, and the reason is structural, not a rule to be memorised: the numerator counts a part of what the denominator counts, so the fraction lies between 0 and 1 inclusive.
• "0.5 means I do not know." It is a claim about the event — the two outcomes are equally likely. Not knowing is a state of the person, and it has no number.
• "Probability 1 means it has always happened before." It means every outcome in the list is one that counts. The all-red bag of sweets is certain because nothing else is in the bag, not because of any history.
• "The five words are five values." Three of them pin a number — impossible is 0, even chance is 0.5, certain is 1. Less likely and more likely are regions, and Fig. 7.1 places 1/6 and 5/6 in them without claiming those are the values.
• "More likely means it will happen." 5/6 is more likely than not and still loses once in six.
• "The scale is a different thing from a fraction." They are the same number. The scale only shows where a fraction between 0 and 1 sits, which is why the chapter compares it to a number line.
• "Bigger deck, bigger probability." In Fig. 7.1 the deck size is fixed at six on purpose. It is the purple count that changes. Anyone who changes both at once has learned nothing from the figure.

Questions to check understanding

• Convert between a probability given as a fraction, a decimal and a percentage
• Given a deck or bag composition, state the probability of drawing a named colour and mark it on a 0-to-1 scale
• Given a probability, say whether the event is more likely than not, and by how much
• Rank several events on a 0-to-1 line and justify each placement — the form Exercise Set 7.1 Q1 takes, where the reasons carry the marks
• Fill in the blanks on the probability of an impossible event and of a certain event — End-of-Chapter Q1 (i) and (iii) (p. 169–170)
• Explain why a probability cannot be greater than 1
• Given a fixed collection size, list every probability that is attainable — the seven rungs of the six-card ladder

Examples worth working on the board

Inputs, not answers, except where the chapter prints the result itself. Values marked Verified are worked out here or an added reading of a printed page; the chapter prints no answers and this volume has no appended answer key.

• The hockey match at three values (p. 157). 0.75 means a 75% chance, and the chapter's own gloss is that winning is then more likely than not. 0.5 means a 50% chance, and the chapter reads it as winning and losing being equally likely. 0 would mean winning is impossible, and the chapter's illustration of that is winning without playing. 1 would mean the win is certain.
• The chapter's own hedge (p. 157). It says most events fall strictly between 0 and 1. That word strictly is doing work: the endpoints are real values, but they are the exceptional cases.
• Fig. 7.1 (p. 158), read off the printed page. A dark green board. Across the middle runs a white axis with a round dot at each end and three tick marks between them, labelled from left to right: Impossible, Less likely, Even chance, More likely, Certain. Below the axis sit five groups of cards, one under each label, each group a block of six cards — two across and three down. The card backs come in two designs, one purple and one green. Above the axis, at the top left, a fanned pile of mixed purple and green cards; to its right, two single cards set apart, one purple and one green, reading as a key to the two colours. Verified counts, purple cards out of six:
  • under Impossible — 0 purple, 6 green
  • under Less likely — 1 purple, 5 green
  • under Even chance — 3 purple, 3 green
  • under More likely — 5 purple, 1 green
  • under Certain — 6 purple, 0 green
• The probabilities those five decks give. Verified: 0/6 = 0; 1/6 ≈ 0.167; 3/6 = 0.5; 5/6 ≈ 0.833; 6/6 = 1. This is the topic's central table and it is not printed anywhere in the chapter — the figure supplies the pictures and leaves the arithmetic to the reader.
• Where the tick marks actually sit, measured off the printed page. The three interior ticks fall at roughly 0.28, 0.54 and 0.75 of the way along the axis. So the ticks are evenly spread label positions, not plotted values: the deck under Less likely is at 0.167 and its tick is at about 0.28. Do not redraw the figure with the ticks pretending to be the probabilities. Either mark the five decks at their true positions, or keep the labels as regions and drop the ticks.
• The two rungs the figure skips, and section 7's payoff. With six cards there are seven possible decks — 0 through 6 purple — spaced one sixth apart. The figure shows five of them and omits 2 and 4. Verified: the full ladder is 0, 1/6 ≈ 0.167, 2/6 ≈ 0.333, 0.5, 4/6 ≈ 0.667, 5/6 ≈ 0.833, 1. The chapter's prose says the likelihood moves smoothly along the scale as the purple count rises (p. 158); showing all seven decks is what makes that sentence true, and it costs two extra panels.
• The chapter's table of five events (pp. 158–159), with the counts it supplies.
  • a number greater than 6 on a die — impossible, because a die's faces run 1 to 6
  • a 3 turning up on one roll of a standard die — less likely, and the chapter's reason is that one face does carry a 3, so the event is not ruled out. Verified: 1/6 ≈ 0.167
  • heads on one flip of a coin — even chance, the two faces being equally likely. Verified: 1/2 = 0.5
  • drawing any number from 2 to 10 from a 52-card deck — more likely, and the chapter gives the count: 36 of the 52 cards carry a number in that range. Verified: 36/52 = 9/13 ≈ 0.692, and 36 is right because four suits each carry the nine values 2 to 10
  • choosing a red sweet from a bag in which every sweet is red — certain. Verified: 1
• Exercise Set 7.1 Q1 (p. 159). The four events to rank — Monday following Sunday; snow in Mumbai in July; an elephant walking through the classroom today; greeting at least one friend at school tomorrow. This topic's job is the placement on the scale; Probability as a measurement, not a guess carries the reasoning. Verified placements: 1, 0, just above 0, well below 1.
• The Summary's notation (p. 173). The probability of an event E is written P(E), and it satisfies 0 ≤ P(E) ≤ 1. This is the only place in the chapter where the inequality is written as an inequality, and it is the compact form of everything section 4 argues.

Figures to have open

• Fig. 7.1 (p. 158) redrawn and able to be shown moving: the 0-to-1 axis with its five printed labels, and beneath it six-card decks whose purple count can be set. Every label in the printed figure is set inside the artwork — the five words and the axis itself are painted on the board — so rebuild it, do not lift it. Two things must change in the redraw: the decks must sit at their true positions rather than under evenly spread ticks, and all seven decks (0 to 6 purple) should be available so section 7 can complete the chapter's own claim about moving smoothly along the scale.
• A part-of-a-whole panel for section 4: one rectangle, a shaded portion, and the fraction written beside it, with the two degenerate shadings shown. Standard schematic.
• The chapter's five-event table (pp. 158–159) turned into five markers on the axis, each carrying the count the chapter supplies. Redraw as a diagram, not as the printed table.
• A single-tick annotation showing 3/4, 0.75 and 75% as one number. Standard schematic.
• No photograph is needed. The 52-card deck can be an icon carrying the number 36.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 7, §7.1.2 "The Probability Scale" (pp. 157–159), running from its heading on p. 157 through Fig. 7.1 and the two-column event table that ends on p. 159.
• Fig. 7.1 (p. 158), with its five labels and five six-card decks, read on the printed page.
• The event table straddles pp. 158–159: three rows on p. 158, two on p. 159.
• Exercise Set 7.1 Q1 (p. 159), shared with Probability as a measurement, not a guess.
• The notation P(E) and the inequality 0 ≤ P(E) ≤ 1 appear only in the Chapter Summary (p. 173); §7.2.2 writes P(Event) and P(Outcome) instead (p. 161).
• End-of-Chapter Q1 (i), (iii) and (iv) (pp. 169–170) test this topic directly.