LEARN · EXPLAIN · REVISE

The probability of each component failing is 1/(k+1), but common power failures can affect both. Can the independent-series formula be used without further evidence?

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

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

TOPIC 01

Independence of events

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

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

01 / Standard#Your turn

k=3

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

Inspect shared causes, not just marginal probabilities.

Worked solution

No; a shared cause may create dependence.

Checks and common pitfalls: A numerical marginal rate does not establish independence.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

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

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.