Theoretical probability: counting favourable outcomes when all are equally likely

What to assume they know

• The 0-to-1 scale, and what the endpoints mean — the 0-to-1 scale, and what the endpoints mean
• Experimental probability: relative frequency over many trials — outcome and sample space, and the experimental route this one is set against
• Counting the members of a small collection, including a collection with repeats
• Simplifying a fraction, and converting it to a decimal and a percentage
• Why long division must either stop or loop — why 1 ÷ 6 gives a non-terminating decimal, so 0.1666… is expected

What they should be able to do

• State the theoretical-probability formula and the assumption it requires
• Explain what "equally likely" demands, and give symmetry as the usual reason it holds
• Compute the theoretical probability of a named outcome for a die, a coin, a set of cards and a spinner
• Choose the sample space so that its elements are equally likely, and explain why the choice matters, using the letters of a word
• Decide, for a described experiment, whether its outcomes are equally likely, and justify the decision
• Distinguish an experiment whose outcomes are equally likely from one whose categories are not, using the marble bag
• Use the chapter's notation P(Event) and P(Outcome) correctly
• Compare the experimental and theoretical routes on what each assumes and what each needs

Where it usually goes wrong

• "Theoretical means true, experimental means approximate." Theoretical is exactly right about an idealisation. Whether the idealisation is the object in front of you is a separate question, and no amount of theory answers it.
• "Count the different things that can happen." PROBABILITY has nine different letters and eleven equally likely positions. Counting the wrong one gives 1/9 instead of 2/11.
• "Two possible results, so each has probability a half." The car either starts or does not. End-of-Chapter Q3 (i) is in the book to kill this.
• "Colours are outcomes." Where a bag holds 3 red marbles against 7 blue ones, the ten marbles are the equally likely outcomes; red and blue are groups of unequal size.
• "A baby is a boy or a girl, so 1/2 exactly." Nearly, and not exactly, and this chapter gives you no evidence either way. The honest answer names the assumption.
• "The formula is the definition of probability." It is a method that works under a stated condition. When the condition fails — the cup of Exercise Set 7.2 Q4 — the formula has nothing to say and the experimental route takes over.
• "P is something you multiply by." P(Event) is notation naming a number, the way sin θ names one. Nothing is being multiplied.
• "A spinner with eight coloured sectors is fair because the colours are different." It is fair, if it is, because the sectors are equal and the spin is unbiased. Colour is decoration.

Questions to check understanding

• Compute a theoretical probability from a described experiment and express it as a fraction, a decimal and a percentage
• Decide whether the outcomes of a described experiment are equally likely, with a reason — the form End-of-Chapter Q3 takes, where the explanation carries the marks
• Compute the probability of drawing a named letter from the letters of a given word
• Compute a probability for a spinner or a numbered-card draw, including a case whose answer is 1
• Compute a probability with a compound condition, such as an even number, or a number above a given value
• Given an experimental value and a theoretical value for the same event, say which is which and explain the gap
• Reasoning question: two students count the outcomes of the same experiment differently and get different answers. Whose sample space has equally likely elements?

Examples worth working on the board

Inputs, not answers, except where the chapter prints the result itself. Values marked Verified are worked out here; the chapter prints no answers and this volume has no appended answer key.

• What §7.2.2 claims (p. 161). Theoretical probability describes what to expect in an ideal, perfectly fair situation; no experiment and no data are needed; it is written P(Event) or P(Outcome). The two "no"s are the section's selling point and its exposure at the same time — nothing is needed except the assumption, and the assumption is everything.
• The formula (p. 161). The count of favourable outcomes over the count of possible outcomes.
• Example 3 (p. 161). A standard 6-sided die, probability of a 4. Favourable count 1 — only the face marked 4. Possible count 6 — the numbers 1 through 6. The chapter prints 1/6 = 0.1666… ≈ 0.167, or 16.7%. Verified. Section 4's job is the sentence the chapter does not write: the 6 in the denominator is not a fact about counting faces, it is the claim that the six faces are interchangeable. On a die with a weight glued inside the 6 is still there and the answer is no longer 1/6.
• Example 4 (pp. 161–162). One letter is taken at random out of the word PROBABILITY, and the chance that it is a B is wanted. The chapter gives favourable count 2, since there are two Bs, and possible count 11, being the number of letters in the word, and prints 2/11 = 0.1818… ≈ 0.182, or 18.2%. Verified letter census: P 1, R 1, O 1, B 2, A 1, I 2, L 1, T 1, Y 1 — eleven letters, nine of them distinct. So the same word gives P(I) = 2/11 and P(a vowel, counting O, A, I, I) = 4/11. Verified.
• The trap section 6 exists for. If you count the nine different letters you get 1/9 ≈ 0.111 for B, and it is wrong. The eleven positions are equally likely because the pick is made from the letters as printed; the nine distinct letters are not, because B and I each occupy two positions. The chapter's careful phrase — the number of letters in the word — is what fixes it, and it deserves one slow beat.
• End-of-Chapter Q3 (p. 170) — the chapter's own test, and the richest question in the chapter for this topic. Five experiments; which have equally likely outcomes, and why? (i) a driver tries the ignition, and the car either starts or fails to; (ii) one toss of a fair coin; (iii) one roll of a fair die with six faces; (iv) a marble taken blind out of a bag in which 3 are red and 7 are blue; (v) the sex of a newborn baby. Verified analysis:
  • (ii) and (iii) — yes, and the reason is symmetry, stated for the coin in the box on p. 164 and never stated for the die.
  • (i) — no. A car in working order starts almost always; the two described results are not two symmetric halves of anything. Nothing about the situation licenses 1/2.
  • (iv) — the question is ambiguous and that is the point. The ten marbles are equally likely; the two colours are not, and come out 3/10 and 7/10. So the answer depends on what you are calling an outcome, which is the topic's thesis showing up in an exercise.
  • (v) — not exactly, and the chapter gives you nothing to decide with. Treat it as very nearly equal and say the chapter supplies no data either way; recorded birth ratios are not quite 1 : 1. Do not present it as an exact fifty-fifty and do not present it as a trick either.
• Exercise Set 7.2 Q5 (p. 165). Probability of an even number on a fair 6-sided die. Verified: favourable {2, 4, 6}, so 3/6 = 1/2. First question in the chapter where the event has more than one favourable outcome.
• Exercise Set 7.2 Q6 (ii) (p. 166). The theoretical probability of a 3 on a die, set against the experimental 3/12. Verified: 1/6 ≈ 0.167. Full treatment in Experimental probability: relative frequency over many trials.
• End-of-Chapter Q4 (p. 170), the parts that are pure counting. (ii) ten identical cards numbered 1 to 10, one drawn, an even number wanted — verified 5/10 = 1/2; (iii) one roll of a die, a number above 4 wanted — verified 2/6 = 1/3; (iv) a bag of 3 red, 2 blue and 1 green ball, one picked, not red wanted — verified 3/6 = 1/2. Note the word identical in (ii): that is the symmetry assumption written into the question, and it is worth pointing at.
• End-of-Chapter Q5 (p. 170). Three candies — strawberry, lemon, mint — one picked at random. Verified: 1/3. The simplest possible instance, and useful because here the three categories and the three objects coincide, which is exactly what fails in Q3 (iv).
• End-of-Chapter Q8 (p. 171). The five letters of PEACE on five cards, one drawn without looking. The page prints the five cards in a row. Verified: the letter census is P 1, E 2, A 1, C 1, so (i) P, E or C gives 1 + 2 + 1 = 4 of 5, and (ii) not an E gives 3 of 5. The same repeated-letter structure as Example 4, one letter shorter, which makes it the natural drilling question.
• End-of-Chapter Q9 (p. 171), the spinner. An arrow is spun and stops with its tip on one of the eight numbers 1 to 8, which the question declares equally likely. Fig. 7.7, read on the printed page: a mounted wheel cut into eight equal sectors, each a different colour, numbered 1 to 8 running clockwise from just right of the top, with a fixed marker at the top of the frame and a black arrow at the hub drawn at rest over sector 2. Verified: (i) 1/8; (ii) odd numbers {1, 3, 5, 7} give 4/8 = 1/2; (iii) above 2 gives {3,…,8} = 6/8 = 3/4; (iv) below 9 gives all eight, so 8/8 = 1 — a certain event; (v) multiples of 3 are {3, 6}, so 2/8 = 1/4. Note that the question supplies the equal-likelihood assumption rather than asking the student to read it off the drawing, which is honest, since the eight sectors are drawn equal but a spinner's fairness is a physical matter.
• End-of-Chapter Q12, the parts that are counting problems (p. 172). (i) two dice rolled, the sum a prime above 5 — verified: the qualifying sums are 7 and 11, reachable in 6 and 2 ways out of 36, so 8/36 = 2/9; (iv) a four-digit number built from 1, 2, 3, 4 with no repeats, the number even — verified: 24 arrangements, even exactly when the last digit is 2 or 4, so 12/24 = 1/2; (v) three multiple-choice questions with four options each, exactly 2 guessed right — verified: 3 × (1/4)² × (3/4) = 9/64 ≈ 0.141. These three are starred in the book and are the stretch band for this topic.
• The summary paragraph (p. 163). Experimental probability rests on collected data and not on assumptions; theoretical probability rests on the equally likely assumption and uses no data. Section 11 is that sentence turned into a two-column comparison, with a third row for what each one cannot do.

Figures to have open

• A die drawn as six separable identical faces, with a second version carrying a hidden weight. Not in the book, and it is what turns the denominator into an argument.
• PROBABILITY set as eleven separate letter tiles that can be highlighted individually, plus a second row showing only the nine distinct letters. Not in the book; the chapter prints the word in prose only.
• Fig. 7.7 (p. 171) redrawn: eight equal sectors numbered 1 to 8 clockwise, a fixed marker at the top, an arrow at the hub. Read on the printed page. All eight numbers are set inside the artwork. Keep the sectors visibly equal — the equal-likelihood claim is the question's premise.
• The five cards of PEACE in a row (End-of-Chapter Q8, p. 171). The chapter prints them as a row of five boxed letters; redraw.
• A ten-marble panel for section 8 in which the marbles can be shown individually and then grouped by colour. Not in the book.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 7, §7.2.2 "Theoretical Probability" (pp. 161–162), comprising the definition and formula on p. 161, Example 3 on p. 161 and Example 4 straddling pp. 161–162.
• §7.2's second numbered route (p. 159), which is where the phrase about no outcome being more likely than another is printed.
• The summarising paragraph comparing the two routes (p. 163), used for section 11.
• The FAIR AND UNBIASED box (p. 164) supplies the symmetry argument section 2 needs; the box itself belongs to Fair, unbiased, and memoryless: the gambler's fallacy.
• Exercise Set 7.2, Q5 and Q6 (ii) (pp. 165–166).
• End-of-Chapter Q3, Q4 (ii)–(iv), Q5 (p. 170), Q8 and Q9 with Fig. 7.7 (p. 171), and the starred Q12 (i), (iv), (v) (p. 172).
• The Chapter Summary (p. 173) restates the formula with the letter A for the event, where §7.2.2 writes P(Event); both name the same thing.