LEARN · EXPLAIN · REVISE

Two independent components each work with probability k/(k+1). A series system needs both. Find success probability.

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#Worked example

k=2

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

Series success is an intersection.

Worked solution

The requested value is 0.44444444.

Checks and common pitfalls: “At least one” would describe a different system.

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.