129i - Comparing Simulated and Theoretical Probabilities

Lesson 129 — Comparing Simulated and Theoretical Probabilities

Strand: Probability | Descriptor: AC9M8P03 | Duration: 45 minutes

Learning Intentions

• To compare experimental and simulated relative frequencies with theoretical probabilities, describing the size of any difference.
• To explain, using the law of large numbers, why relative frequency approaches theoretical probability as the number of trials increases.

Success Criteria

I can:

1. Calculate the absolute and proportional difference between an observed result and a theoretical probability.
2. State what the law of large numbers predicts about relative frequency as the number of trials increases.
3. Identify plausible sources of variation between a prediction and an observed result: ordinary chance, bias, or a wrong model.
4. Judge, using data at different trial sizes, whether a difference is ordinary chance variation or points to a genuine problem.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. Expected frequency ? (formula)
2. Absolute difference ? (formula)
3. A coin gives heads in tosses. Find the absolute difference from the expected frequency.
4. Find the relative frequency for Q3, and its difference from the theoretical probability .

Answers: 1. ; 2. observed predicted; 3. expected , difference ; 4. relative frequency , difference .

Activities

Activity 1 — Explicit Instruction: the Law of Large Numbers (14 min)

I do / We do / You do.

I do: A simulated fair coin is tossed increasingly many times.

Trials ()	Heads	Relative frequency

Say aloud: “As grows, the relative frequency gets closer to , and the swings shrink.” This is the law of large numbers: as the number of trials increases, relative frequency tends to settle closer to the true theoretical probability.

We do: Together compute the absolute and proportional difference from at and .

Notice: the absolute difference actually grew (), but the proportional difference shrank sharply (). It is the proportional difference — the relative frequency itself — that the law of large numbers describes.

You do: A die’s number of s is recorded at increasing trial sizes: sixes; sixes; sixes. Calculate each relative frequency and its difference from . Describe the trend.

(Answers: (); (); () — the relative frequency converges steadily toward the theoretical value as grows.)

Activity 2 — Applied Task: Sources of Variation (20 min)

Pairs, then whole-class share.

For each scenario, judge the most likely explanation: ordinary chance variation, likely bias (the device itself is unfair), or likely wrong model (the assumed probabilities were incorrect from the start). Justify each judgement.

Scenario A. A coin gives heads in tosses. Scenario B. The same type of coin gives heads in tosses. Scenario C. A spinner assumed to have equal sectors gives results wildly skewed toward one sector across spins, and on close inspection that sector is visibly larger than the others. Scenario D. A die rolled times gives relative frequencies for each face all within of .

Socratic scaffolding for comparing A and B (the harder part):

Prompt	Purpose
What is the same about A and B?	Both give a relative frequency of — the same proportional gap from ().
What is different?	B used times as many trials as A.
What would you expect a fair coin’s gap to do as trials increase?	Shrink, following the law of large numbers, as in Activity 1.
So what does it mean that the gap has not shrunk in B?	A gap that persists — rather than shrinking — as the sample grows is no longer well explained by ordinary chance; it points toward the coin itself being biased.
Looking back — restate the judgement for each.	A: too few trials to judge, plausibly ordinary chance. B: the same gap sustained over the trials is good evidence of bias.

Answers: A — ordinary chance variation; trials is too few to draw a conclusion from an gap. B — likely bias; the same proportional gap persisting at the trials, where the law of large numbers predicts it should shrink, is meaningful evidence. C — likely wrong model; the sectors were never actually equal, so the assumed theoretical probabilities ( each) were wrong from the outset, independent of any chance variation. D — no evidence of a problem; every face sits close to theoretical, exactly as expected for a fair die with many trials.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. State the law of large numbers in one sentence.
2. A die gives a relative frequency of s equal to after rolls, and after rolls. Which estimate is more reliable, and why?
3. A coin gives close to heads consistently across several independent samples of tosses each. Is this more likely ordinary variation or bias? Explain.
4. Name the three possible explanations for an observed result differing from a prediction.

Answers: 1. As the number of trials increases, relative frequency tends to settle closer to the true theoretical probability. 2. The -roll estimate, since larger samples are less affected by chance swings, following the law of large numbers. 3. Bias — a persistent, sizeable gap repeated across several large, independent samples is not well explained by ordinary chance, which should shrink such gaps as trials increase. 4. Ordinary chance variation; bias in the device/procedure; a wrong model (incorrect assumed probabilities).

Common Misconceptions

Misconception	How to pre-empt it
Believing the law of large numbers means future trials “correct” or “cancel out” an earlier imbalance.	Clarify: the proportion of that imbalance shrinks because it is diluted by more trials, not because it is undone — Activity 1’s absolute-vs-proportional contrast makes this explicit.
Assuming more trials guarantee an exact match to the theoretical probability.	Even at in Activity 1, the relative frequency is close but not exactly .
Judging any deviation, at any sample size, as evidence of bias.	Contrast Scenario A (too few trials) directly against Scenario B (same gap, far more trials).
Confusing absolute difference with proportional difference when judging significance.	Revisit Activity 1’s key observation: absolute difference can grow even as proportional difference shrinks.
Treating “wrong model” and “bias” as the same explanation.	Scenario C (a visibly uneven spinner) is a wrong-model case from the start, not a fair device behaving unfairly by chance.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A die is rolled times, giving sixes. Find both the absolute and proportional differences from the expected result, and judge whether this suggests a problem.

Answer

Expected . Absolute difference . Proportional: . A small proportional gap over many trials — consistent with ordinary variation for a fair die.

E2 (Kangaroo style). In trials, an event’s relative frequency was . In a second, independent run of trials with the same equipment, the relative frequency was . What does this comparison suggest about the theoretical probability?

Answer

The two proportional results are almost identical despite the second sample being larger — this stability across a much larger sample is strong evidence the true probability really is close to , rather than the first result being a lucky/unlucky fluke.

E3 (Challenge). A relative frequency of is measured after trials of a coin known to be fair, with absolute difference from the expected frequency equal to (a typical-sized swing). Find the proportional difference in terms of , and evaluate it at , and . What do you notice?

Answer

At : . At : . At : . The proportional difference shrinks steadily as grows — a direct algebraic illustration of the law of large numbers.

Homework

1. State, in your own words, what the law of large numbers predicts.
2. A spinner gives relative frequency for its largest sector after spins, and after spins (theoretical probability ). Which estimate should be trusted more, and why?
3. Find the absolute and proportional difference from the expected frequency: a die rolled times gives sixes.
4. Explain why a persistent gap across several large samples is stronger evidence of bias than a single small sample’s gap.
5. Reasoning. A friend says: “After tosses my coin has given exactly heads every time I check the running total at multiples of , so it must be broken.” Explain what is wrong with this reasoning.
6. Reasoning. Give one example each of ordinary chance variation, bias, and a wrong model, different from those used in this lesson.
7. Challenge. A biased coin’s true probability of heads is . After tosses, the observed relative frequency is , and after tosses (same coin, same procedure) it is still . Explain, referring to the law of large numbers, what this pair of results implies about .

Answers: 1. Student’s own — relative frequency tends to settle closer to the theoretical probability as the number of trials increases. 2. The -spin estimate, since it is far less affected by chance swings. 3. Expected ; absolute ; proportional . 4. A single small sample’s gap is easily produced by ordinary chance; the same gap repeating across several large samples is very unlikely to occur by chance alone if the true probability were as assumed, so bias becomes the better explanation. 5. Checking a running total at intervals doesn’t create new evidence — a coin’s relative frequency naturally fluctuates around ; without comparing to the expected fluctuation size, “exactly ” observed once is not itself remarkable, and describing the coin as “broken” is not justified by this alone. 6. Student’s own, e.g. ordinary variation — a coin giving heads in ; bias — a die consistently favouring one face across thousands of rolls; wrong model — assuming a spinner’s sectors are equal when a manufacturing flaw makes one larger. 7. Since the gap of from has not shrunk despite more trials, the law of large numbers suggests this is not ordinary chance variation settling down — it is likely that genuinely differs from , close to .