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Two draws from a finite bag are made without replacement. Which exact count model is appropriate?

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

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

Revisit first: Conditions, partitions and posterior probabilityDistribution means, variances and dependence

TOPIC 01

Choosing binomial, hypergeometric and normal models

Choose a model from its assumptions and support before computing its probability or moments.

What you will be able to explain

  • Choose a model from its assumptions and support before computing its probability or moments.
  • Handle census and deterministic endpoints; standardize continuous normal variables.

Try it. Leave your reasoning visible.

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

01 / Foundation#Your turn

Two draws from a finite bag are made without replacement. Which exact count model is appropriate?

  • Draw uniformly from the remaining finite bag without replacement, except the explicitly compared replacement model.
Skills and prerequisite lessons

Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.

  1. Binomial without checking conditions
  2. Hypergeometric
  3. Normal exactly
  4. Always uniform over counts
Working and explanation
BUILD THE REASONING
Hint 1
Track the finite population without replacement.
Hint 2
Start from this intermediate relation.
P(X=k)=(Kk)(N−Kn−k)/(Nn)P(X=k)=\binom Kk\binom{N-K}{n-k}/\binom Nn
Worked solution
  1. Track the finite population without replacement.
  2. Establish and simplify this mathematical relation.
    P(X=k)=(Kk)(N−Kn−k)/(Nn)P(X=k)=\binom Kk\binom{N-K}{n-k}/\binom Nn
  3. Establish and simplify this mathematical relation.
    max⁡(0,n−N+K)≤k≤min⁡(n,K)\max(0,n-N+K)\le k\le\min(n,K)
  4. Individual samples may be equally likely while success counts are not.
Hypergeometric
Checks and common pitfalls: Individual samples may be equally likely while success counts are not.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5PredictPredict whether replacement increases the two-red probability from a mostly-red finite bag.
8Build the ideaSample response: without replacement the success fraction drops after a red draw.
10Guided comparisonSample response: 9/25 with replacement, 3/10 without; enumerate pairs to check.
12Independent practice and discussionSort scenarios by assumptions and use integer checks for reliability.
5Exit and next stepExit: when drawing all six objects with two reds, X=2 surely and variance is zero.
Question sequence
  • Predict whether replacement increases the two-red probability from a mostly-red finite bag.
  • Sample response: 9/25 with replacement, 3/10 without; enumerate pairs to check.
Board development
  • Mechanism determines model
    P(X=k)=(nk)pk(1−p)n−kP(X=k)=\binom nkp^k(1-p)^{n-k}
Expected correct and incorrect responses
  • Correct sample: Hypergeometric
  • Anticipated wrong response: Binomial without checking conditions — Uses fixed trial count alone to justify a binomial model.
  • Anticipated wrong response: 9 — Reads the normal variance parameter as standard deviation.
  • Choosing binomial from fixed n alone.
  • Using normal variance as its standard deviation.
Follow-up questions
  • Follow-up: Track the finite population without replacement. Then explain why Binomial without checking conditions fails.
  • Follow-up: Read the second parameter as variance. Then explain why 9 fails.
  • Set n=N and explain why only K successes are possible without replacement.
Remediation
  • Sample response: without replacement the success fraction drops after a red draw.
  • Return to the first example, identify the missing condition, then solve one fresh task without viewing its solution.
Exit check
  • Exit: when drawing all six objects with two reds, X=2 surely and variance is zero.
Assessment checklist and sample responses
  • Representation and conditions
    No replacement changes the next success fraction, so finite count is hypergeometric.
  • Connected mathematical reasoning
    Two red probability from three red of five is (3/5)(2/4)=3/10.
  • Interpretation and boundary check
    Drawing the full bag with two reds gives X=2 surely and variance zero.
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Curriculum and source notes ↗