LEARN · EXPLAIN · REVISE

Independent trials succeed with probability 1/2. Find the least n for success at least once with probability ≥1−2^(−m).

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

高三選擇性必修第三册(A版).pdf · 7.4 · PDF 77 / printed page 72

TOPIC 01

Binomial and hypergeometric distributions

Choose the correct model for repeated independent trials or sampling without replacement.

Choose the correct model for repeated independent trials or sampling without replacement.
Justify the method and check the conditions in a new situation.

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

01 / Transfer#Your turn

m=6

Binomial trials have fixed n, common p and independence; hypergeometric sampling uses a finite population without replacement.

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

Use this intermediate relation.

1-2^{-n}\ge1-2^{-m}

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$2^{-n}\le2^{-6}\Rightarrow n\ge6$
Use the complementary event of no successes.

The requested value is 6.

Checks and common pitfalls: Use the complementary event of no successes.

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

Teacher preparation and assessment

Choose the correct model for repeated independent trials or sampling without replacement.
Which condition is essential in binomial and hypergeometric distributions?
Which probability changes after one success is removed from a finite bag?

Defining relation: Choose the correct model for repeated independent trials or sampling without replacement.
$P(X=k)=C_n^kp^k(1-p)^{n-k}$
Conditions: Binomial trials have fixed n, common p and independence; hypergeometric sampling uses a finite population without replacement.