Kenneth ChengMathematical modeling
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A,B are independent and P(B)>0. Compare P(A|B) with P(A).

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

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高一必修 第二册(A版).pdf · 10.2 · PDF 256 / printed page 249

Revisit first: Random events and probability

TOPIC 01

Independence of events

Build understanding of independence of events through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use independence of events 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 / Foundation#Your turn

A,B are independent and P(B)>0. Compare P(A|B) with P(A).

P(B)=1/4P(B)=1/4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use independence only when justified and keep it distinct from disjointness.
Hint 2
Use the product definition of independence.
Worked solution

  1. Use independence only when justified and keep it distinct from disjointness.

  2. Calculate or simplify this relation.

    P(A∣B)=P(A∩B)/P(B)=P(A)P(A|B)=P(A\cap B)/P(B)=P(A)

  3. Conditioning on B does not change A’s probability in this model.

They are equal.

Checks and common pitfalls: Conditioning on B does not change A’s probability in this model.

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 independence of events?
  • Which representation makes this task easier, and why?
  • Change one assumption. Does the conclusion survive?

Board plan

  • Use independence of events to state a justified conclusion.
  • Independence: Independent events satisfy the product rule.
    P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B)
  • Model check: Shared causes and sampling without replacement can create dependence.
  • Close with: conditions → representation → reasoning → check.

Anticipated thinking

  • Expected reasoning: Independent events satisfy the product rule.
  • Expected reasoning: Shared causes and sampling without replacement can create dependence.
  • Expected correction: Known marginal probabilities do not establish independence.

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 ↗