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

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

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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.

Try it. Leave your reasoning visible.

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

01 / 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.

Focus on one question

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.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗