Kenneth ChengMathematical modeling
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Frequency and probability

Read the idea, work independently, then explain what changed.

高一必修 第二册(A版).pdf · 10.3 · PDF 261 / printed page 254

Revisit first: Independence of events

TOPIC 01

Frequency and probability

Build understanding of frequency and probability through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use frequency and probability to state a justified conclusion.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Relative frequency

The observed fraction estimates a model probability over repeated trials.

Finite-sample uncertainty

A long-run tendency does not force exact agreement in every finite sample.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in frequency and probability changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Binomial model: fixed n=5, independent trials, common p=0.5. E(X)=2.5, Var(X)=1.25. Bars show exact probabilities.

Binomial model: fixed n=5, independent trials, common p=0.5. E(X)=2.5, Var(X)=1.25. Bars show exact probabilities.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

An event occurs 3k times in 10k trials. Find its relative frequency.

k=2k=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Use occurrences divided by trials.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    f=6/20=0.3f=6/20=0.3

  3. Observed frequency is data, not an exact guarantee of future outcomes.

The requested value is 0.3.

Checks and common pitfalls: Observed frequency is data, not an exact guarantee of future outcomes.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Two batches have 2k successes in 5k trials and 3k in 10k. Find pooled relative frequency.

k=2k=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Pool counts and trial totals before dividing.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    f=(4+6)/(10+20)=1/3f=(4+6)/(10+20)=1/3

  3. A simple average of batch percentages ignores their unequal sizes.

The requested value is 0.33333333.

Checks and common pitfalls: A simple average of batch percentages ignores their unequal sizes.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

In k trials, the event never occurs. Must its true probability be zero? Explain.

k=2k=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Distinguish a finite observation from a universal model claim.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. For a counterexample, choose k independent Bernoulli trials with the same event probability 0<p<1.

  3. Calculate or simplify this relation.

    P(zero occurrences)=(1−p)k>0(0<p<1)P(\text{zero occurrences})=(1-p)^k>0\quad(0<p<1)

  4. A finite sample cannot establish impossibility merely through non-occurrence.

No. A positive-probability event can be absent in a finite sample.

Checks and common pitfalls: A finite sample cannot establish impossibility merely through non-occurrence.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

An event occurs 3k times in 10k trials. Find its relative frequency.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Use occurrences divided by trials.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    f=9/30=0.3f=9/30=0.3

  3. Observed frequency is data, not an exact guarantee of future outcomes.

The requested value is 0.3.

Checks and common pitfalls: Observed frequency is data, not an exact guarantee of future outcomes.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Two batches have 2k successes in 5k trials and 3k in 10k. Find pooled relative frequency.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Pool counts and trial totals before dividing.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    f=(6+9)/(15+30)=1/3f=(6+9)/(15+30)=1/3

  3. A simple average of batch percentages ignores their unequal sizes.

The requested value is 0.33333333.

Checks and common pitfalls: A simple average of batch percentages ignores their unequal sizes.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

A model has p=1/4. In 20k independent trials, find the expected number of occurrences.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Expected count equals trial number times model probability.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    E(N)=np=60/4=15E(N)=np=60/4=15

  3. The actual integer count need not equal its expectation.

The requested value is 15.

Checks and common pitfalls: The actual integer count need not equal its expectation.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

After k consecutive heads from an independent fair coin, what is the probability the next toss is heads?

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Independence keeps the next-trial probability unchanged.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    P(Hnext∣H1,…,Hk)=1/2P(H_{next}|H_1,…,H_k)=1/2

  3. The coin does not compensate for previous outcomes.

The requested value is 0.5.

Checks and common pitfalls: The coin does not compensate for previous outcomes.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

A simulation makes 10k equally likely selections from labels 1,…,k. What model probability does label 1 have?

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Separate the per-trial model from the number of simulations.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    p=1/3p=1/3

  3. More simulations improve empirical evidence; they do not alter the specified probability.

The requested value is 0.33333333.

Checks and common pitfalls: More simulations improve empirical evidence; they do not alter the specified probability.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

A program’s pseudo-random outputs are reused with the same seed. Does that create new independent evidence each rerun?

n=30n=30
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Inspect whether the sequence actually changes.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    same seed⇒same deterministic sequence\text{same seed}\Rightarrow\text{same deterministic sequence}

  3. Reproducibility is valuable but does not multiply the number of independent runs.

No; reproducing the same sequence repeats the same simulated observations.

Checks and common pitfalls: Reproducibility is valuable but does not multiply the number of independent runs.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

In k trials, the event never occurs. Must its true probability be zero? Explain.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Distinguish a finite observation from a universal model claim.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. For a counterexample, choose k independent Bernoulli trials with the same event probability 0<p<1.

  3. Calculate or simplify this relation.

    P(zero occurrences)=(1−p)k>0(0<p<1)P(\text{zero occurrences})=(1-p)^k>0\quad(0<p<1)

  4. A finite sample cannot establish impossibility merely through non-occurrence.

No. A positive-probability event can be absent in a finite sample.

Checks and common pitfalls: A finite sample cannot establish impossibility merely through non-occurrence.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

An event occurs 3k times in 10k trials. Find its relative frequency.

k=4k=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Use occurrences divided by trials.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    f=12/40=0.3f=12/40=0.3

  3. Observed frequency is data, not an exact guarantee of future outcomes.

The requested value is 0.3.

Checks and common pitfalls: Observed frequency is data, not an exact guarantee of future outcomes.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Two batches have 2k successes in 5k trials and 3k in 10k. Find pooled relative frequency.

k=4k=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Pool counts and trial totals before dividing.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    f=(8+12)/(20+40)=1/3f=(8+12)/(20+40)=1/3

  3. A simple average of batch percentages ignores their unequal sizes.

The requested value is 0.33333333.

Checks and common pitfalls: A simple average of batch percentages ignores their unequal sizes.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

A model has p=1/4. In 20k independent trials, find the expected number of occurrences.

k=4k=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish observed relative frequency from a model probability and describe the experiment.
Hint 2
Expected count equals trial number times model probability.
Worked solution

  1. Distinguish observed relative frequency from a model probability and describe the experiment.

  2. Calculate or simplify this relation.

    E(N)=np=80/4=20E(N)=np=80/4=20

  3. The actual integer count need not equal its expectation.

The requested value is 20.

Checks and common pitfalls: The actual integer count need not equal its expectation.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for frequency and probability?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use frequency and probability to state a justified conclusion.
    • Relative frequency: The observed fraction estimates a model probability over repeated trials.
    • Finite-sample uncertainty: A long-run tendency does not force exact agreement in every finite sample.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: The observed fraction estimates a model probability over repeated trials.
    • Expected reasoning: A long-run tendency does not force exact agreement in every finite sample.
    • Expected correction: Previous independent outcomes do not create compensation on the next trial.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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