LEARN · EXPLAIN · REVISE

A bag has k red and 1 blue; draw twice without replacement. Are the two red-draw events independent?

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

Update the bag after the first outcome.

Worked solution

No; P(second red|first red)=(k−1)/k differs from k/(k+1).

Checks and common pitfalls: Without replacement, the first draw changes the second distribution.

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.