Experimental probability: relative frequency over many trials

What to assume they know

• What we mean by random — random experiments, trials, and repeatability
• The 0-to-1 scale, and what the endpoints mean — the 0-to-1 scale
• Writing a count over a total as a fraction, and converting it to a decimal and a percentage
• Why long division must either stop or loop — why a division such as 1 ÷ 6 need not terminate, so that 0.1666… does not look like a mistake
• Reading a small frequency table and totalling its rows

What they should be able to do

• State the two objective routes §7.2 offers, and say what evidence each one uses
• Define outcome and sample space as §7.2.1 introduces them, and write the sample space for a coin and for a die
• State the experimental-probability formula and identify what each part of it counts
• Compute an experimental probability from a stated number of trials and successes, as a fraction, a decimal and a percentage
• Explain that relative frequency and experimental probability are two names for the same number
• Explain why an experimental value need not match a theoretical one, and what changes as the number of trials grows
• State the Law of Large Numbers, distinguishing what it promises about proportions from what it does not promise about counts
• Explain why an experiment is the only route available for an object with no symmetry, such as a paper cup
• Carry out a 20-trial coin experiment, record the results, and compute the experimental probability from your own record
• Distinguish a frequency from a relative frequency in a survey table

Where it usually goes wrong

• "The experiment gives you the true probability." It gives an estimate. The chapter's own Example 2 estimates 1/6 as 0.16.
• "If it does not come out 1/6, the die must be loaded." Fifty rolls of a fair die will usually not give exactly 1/6 — it cannot, since 1/6 of 50 is not a whole number. The instrument is coarse, not the die crooked.
• "More trials makes each individual trial more predictable." It makes the proportion more stable. The next roll is exactly as unpredictable as the first.
• "The Law of Large Numbers says the counts even out." It says the ratio settles. The absolute gap between heads and tails can and typically does keep growing. This is the single most consequential misreading in the chapter and it is what Fair, unbiased, and memoryless: the gambler's fallacy is about.
• "Relative frequency is something different from experimental probability." They are the same quotient; the chapter says so in the same breath as Example 2.
• "An experiment is what you do when you are too lazy to work out the theory." For a paper cup there is no theory to work out.
• "15 is the relative frequency." 15 is a count. The relative frequency is 15 out of 50. End-of-Chapter Q2 exists to catch exactly this swap.

Questions to check understanding

• Given a number of trials and a count of successes, compute the experimental probability as a fraction, a decimal and a percentage
• Given an experimental and a theoretical value, explain why they differ and predict what happens as trials increase — the exact form of Exercise Set 7.2 Q6 (iii)
• Perform a stated experiment for a stated number of trials, tabulate, and compute the experimental probabilities
• Assign probabilities to the outcomes of an asymmetric object from experimental data, and check they total 1
• Identify the frequency and the relative frequency in a survey statement
• State the Law of Large Numbers and say what it does and does not promise
• Reasoning question: a student rolls a die 12 times, gets no sixes at all, and concludes the probability of a six is 0. What is wrong with the conclusion?

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.

• §7.2's two routes (p. 159). The first collects data — either by running an experiment many times, or by analysing statistical data already gathered — and in both cases computes a relative frequency. The second assumes every outcome is equally likely and reasons about counts. The chapter numbers them 1 and 2 and names them experimental and theoretical probability. Worth showing both before opening either, because the whole module is the comparison.
• Outcome and sample space, introduced here (p. 160). In an experiment a result is called an outcome, and the collection of all of them is the sample space, written inside braces with commas between the items. Example 1 on the page: tossing a coin gives {H, T}; rolling a die gives {1, 2, 3, 4, 5, 6}. Figs. 7.2 and 7.3 illustrate the coin and a six-sided die.
• The formula (p. 160). Experimental probability is the number of times the event happened divided by the total number of trials. Both counts come out of the record you kept; nothing is assumed about the object.
• Example 2 (p. 160). A die is rolled 50 times and lands on 4 exactly 8 times. The chapter computes 8/50 = 0.16, or 16%, and then says the same number is the relative frequency of rolling a 4.
• Section 5's arithmetic, which the chapter does not do and which flips how Example 2 reads. Verified: a fair die rolled 50 times should give 50 × 1/6 ≈ 8.33 fours. The attainable whole numbers either side are 8 and 9, giving 8/50 = 0.16 and 9/50 = 0.18. Against the target 1/6 ≈ 0.1667 the errors are 0.0067 and 0.0133 — so 8 is the best count a fifty-roll run could have produced. Example 2 is not an unlucky sample being paraded as a warning; it is as close to right as fifty rolls allows. Say this. It changes the emotional register of the whole section from "experiments are unreliable" to "experiments are limited by their grain".
• Exercise Set 7.2 Q6 (p. 166). A 6-sided die is rolled 12 times and a 3 comes up three times. Asked: the experimental probability, the theoretical probability, why they differ, and what to expect at 60, 600 or 6000 rolls. Verified: experimental 3/12 = 0.25; theoretical 1/6 ≈ 0.167; the gap is 0.083, which is large because twelve rolls is a coarse instrument — the attainable values with 12 rolls are 0, 1/12, 2/12, 3/12, … and 1/6 = 2/12 is attainable, so the run overshot the exact value by a single occurrence rather than falling short of it: two of that face in twelve rolls would have landed on 1/6 exactly, and the die delivered three. (The face at stake here is the 3. The 4 belongs to Example 2 and to the p. 163 box.) At 60 rolls the expected count is 10; at 600 it is 100; at 6000 it is 1000, and the proportion should sit visibly closer to 0.167 each time. Verified.
• The Law of Large Numbers (p. 163). The chapter's statement: even in a perfectly fair set-up the experimental value can differ from the theoretical one, especially when there are few trials, and as trials increase the experimental value tends towards the theoretical. That is the guarantee the whole method rests on, and it is stated once, in the paragraph that summarises §7.2.
• Think and Reflect, p. 163. Having rolled a 4 eight times running, the probability of a 4 next time is unchanged; probability speaks about the long run, not the next trial. Note the printed figure here — see the flag in Notes — and use 1/6 ≈ 0.167.
• Exercise Set 7.2 Q4, the paper cup (p. 165). Throw a paper cup up in the air a hundred times, and after each throw note which of three ways it has come down — standing on its base, inverted on its rim, or lying sideways; then assign probabilities from the record. Fig. 7.5 shows a boy tossing a cup and three landing positions on a table — read on the printed page, they are, left to right: the cup standing upright with its opening up, the cup inverted resting on its rim, and the cup lying on its side. The caption names them bottom, top and side in that order. Verified point for section 9: there is no way to count equally likely faces for a cup, so no theoretical value exists to compare against. This is the only question in the chapter where the experiment is not one of two routes but the only route. And the three assigned values come out totalling 1, because the three counts come out totalling 100 — which is worth pausing on, since the chapter never states anywhere in these nineteen pages that the probabilities over a sample space add to 1. The question does not need that fact (each value is its own count over 100, by the p. 160 formula) but it hands the student a clean instance of it, which is the better teaching opportunity of the two.
• Exercise Set 7.2 Q3 (p. 165). Toss a coin 20 times, record each result, count the heads and the tails, compute the experimental probability of heads, and then state the probability of tails on one further toss. Verified: the first three parts come from the student's own table and will differ from student to student; the fourth is 1/2 whatever the table said, and it is a gambler's-fallacy trap planted in the experimental set — see Fair, unbiased, and memoryless: the gambler's fallacy.
• End-of-Chapter Q2 (p. 170). In a survey of 50 students, 15 said they liked football. The blanks ask which of frequency and relative frequency the number 15 is, and then for the other one as a fraction or decimal. Verified: 15 is the frequency; the relative frequency is 15/50 = 3/10 = 0.3.
• The Did you know? box (p. 161). Snakes and Ladders is traced to an older Indian dice game, Jñān-Chaupaḍ, used as a teaching tool, with each ladder standing for a virtue and each snake for a vice, and versions found in different parts of India. Fig. 7.4 is a photograph of a 19th-century Jain painted-cloth board from the National Museum. Use it as the section-1 or section-12 breather; the game returns as Example 6 on p. 164, owned by Fair, unbiased, and memoryless: the gambler's fallacy.
• End-of-Chapter Q7 (p. 171), borrowed for section 11. A tyre company's record of 1000 cases. Its four columns give: under 4000 km, 20 cases; 4001 to 9000 km, 210; 9001 to 14000 km, 325; over 14000 km, 445. Verified: the counts total 1000, and reading probabilities straight off the table gives 0.02, 0.535 and 0.445, which total 1. The full treatment belongs to Estimating from statistical data, and scaling the estimate up.

Figures to have open

• Fig. 7.2 and Fig. 7.3 (p. 160): a coin showing its two faces, and a six-sided die. Standard schematics; the printed versions are decorative and carry only H and T inside the artwork.
• Fig. 7.5 (p. 165) redrawn: the three paper-cup landing positions — upright, inverted, on its side — clearly distinguished and labelled bottom, top and side to match the caption. Read on the printed page. This figure is essential, because the whole point of the question is that the three positions are not interchangeable and not equally likely.
• A running-proportion chart: trials on the horizontal axis on a log-like spread (12, 60, 600, 6000), the running proportion on the vertical, with a horizontal line at 1/6. Not in the book, and it is what makes section 8 an argument rather than an assertion.
• A single tally sheet that can be filled live, for sections 4 and 10. Standard schematic.
• Fig. 7.4 (p. 161) is a photograph of a museum object and should not be reproduced. Use a redrawn board-game motif with snakes and ladders on it, and name the museum piece when explaining it.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 7, §7.2 "Measuring Probability Objectively" (p. 159) and §7.2.1 "Experimental Probability: Performing Observations or Experiments" (pp. 160–161), together with Figs. 7.2 and 7.3 (p. 160).
• The Did you know? box and Fig. 7.4 (p. 161), which sit between §7.2.1 and §7.2.2.
• The Law of Large Numbers paragraph and the Think and Reflect that follows it (p. 163) — printed at the end of §7.2.3 but summarising all of §7.2, which is why this topic carries them.
• Exercise Set 7.2, Q3, Q4 and Q6 (pp. 165–166), with Fig. 7.5 (p. 165).
• End-of-Chapter Q2 (p. 170); End-of-Chapter Q7 (p. 171) is borrowed for section 11 and belongs to Estimating from statistical data, and scaling the estimate up.
• The Chapter Summary (p. 173) restates the experimental-probability formula and says nothing about the Law of Large Numbers. Checked on the printed page.