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A and B are disjoint and both have positive probability. Can they be independent?

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

高三選擇性必修第三册(A版).pdf · 7.1 · PDF 49 / printed page 44

TOPIC 01

Conditional and total probability

Use conditional denominators, disjoint partitions and Bayes inversion.

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

01 / Standard#Your turn

P(A)=1/15,\quad P(B)=1/15

The conditioning event must have positive probability; a total-probability partition is exhaustive and disjoint.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the event with a conditional probability or a disjoint partition.

Hint 2

Compare the intersection with the product.

Worked solution

The requested relation or conclusion is shown below.

P(A\cap B)=0\ne P(A)P(B)

Checks and common pitfalls: Disjoint positive-probability events are dependent.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Defining relation: Use conditional denominators, disjoint partitions and Bayes inversion.
$P(A|B)=P(A\cap B)/P(B)$
Conditions: The conditioning event must have positive probability; a total-probability partition is exhaustive and disjoint.