LEARN · EXPLAIN · REVISE

After k consecutive heads from an independent fair coin, what is the probability the next toss is heads?

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

高一必修第二册(A版).pdf · 10.3 · PDF 261 / printed page 254

TOPIC 01

Frequency and probability

Build understanding of frequency and probability through definitions, contrasting cases and justified applications.

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

01 / Foundation#Your turn

k=3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish observed relative frequency from a model probability and describe the experiment.

Hint 2

Independence keeps the next-trial probability unchanged.

Worked solution

Distinguish observed relative frequency from a model probability and describe the experiment.
Calculate or simplify this relation.
$P(H_{next}|H_1,…,H_k)=1/2$
The coin does not compensate for previous outcomes.

The requested value is 0.5.

Checks and common pitfalls: The coin does not compensate for previous outcomes.

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

Teacher preparation and assessment

Use frequency and probability to state a justified conclusion.
Relative frequency: The observed fraction estimates a model probability over repeated trials.
Finite-sample uncertainty: A long-run tendency does not force exact agreement in every finite sample.
Close with: conditions → representation → reasoning → check.

Expected reasoning: The observed fraction estimates a model probability over repeated trials.
Expected reasoning: A long-run tendency does not force exact agreement in every finite sample.
Expected correction: Previous independent outcomes do not create compensation on the next trial.