Kenneth ChengMathematical modeling
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Binomial and hypergeometric distributions

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

Revisit first: Numerical characteristics of random variables

TOPIC 01

Binomial and hypergeometric distributions

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

What you will be able to explain

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

Defining relation

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

P(X=k)=Cnkpk(1−p)n−kP(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.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Which probability changes after one success is removed from a finite bag?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Draw n=4 without replacement from ten objects, K=4 successes. Exact hypergeometric probabilities; E(X)=1.6, Var(X)=0.64. Trials are not independent.

Draw n=4 without replacement from ten objects, K=4 successes. Exact hypergeometric probabilities; E(X)=1.6, Var(X)=0.64. Trials are not independent.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Compare with-replacement and without-replacement models using the same bag.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

X~Bin(m,1/2). Find P(X=0).

m=5m=5
  • 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.
P(X=0)=(1−p)mP(X=0)=(1-p)^m
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    P(X=0)=2−5P(X=0)=2^{-5}

  3. Zero successes requires failure in every independent trial.

The requested value is 0.03125.

Checks and common pitfalls: Zero successes requires failure in every independent trial.

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

02 / Standard#Worked example

A bag has t red and 3 blue balls; select 2 without replacement. Find the probability both are blue.

t=3t=3
  • 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.
C32/Ct+32C_3^2/C_{t+3}^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    P=3/15P=3/15

  3. The unordered sample denominator counts pairs, not independent trials.

The requested value is 0.2.

Checks and common pitfalls: The unordered sample denominator counts pairs, not independent trials.

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

03 / Transfer#Worked example

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

m=7m=7
  • 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≥1−2−m1-2^{-n}\ge1-2^{-m}
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    2−n≤2−7⇒n≥72^{-n}\le2^{-7}\Rightarrow n\ge7

  3. Use the complementary event of no successes.

The requested value is 7.

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

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

04 / Foundation#Your turn

X~Bin(m,1/2). Find P(X=0).

m=8m=8
  • 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.
P(X=0)=(1−p)mP(X=0)=(1-p)^m
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    P(X=0)=2−8P(X=0)=2^{-8}

  3. Zero successes requires failure in every independent trial.

The requested value is 0.00390625.

Checks and common pitfalls: Zero successes requires failure in every independent trial.

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

05 / Foundation#Your turn

X~Bin(m,1/2). Find P(X=1).

m=9m=9
  • 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.
P(X=1)=Cm1/2mP(X=1)=C_m^1/2^m
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    P(X=1)=9/512P(X=1)=9/512

  3. There are m possible locations of the single success.

The requested value is 0.017578125.

Checks and common pitfalls: There are m possible locations of the single success.

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

06 / Foundation#Your turn

X~Bin(m,1/4). Find E(X).

m=10m=10
  • 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
List the possible values and probabilities before computing moments.
Hint 2
Use this intermediate relation.
E(X)=mpE(X)=mp
Worked solution

  1. List the possible values and probabilities before computing moments.

  2. Apply the stated relation and retain its conditions.

    E(X)=10/4E(X)=10/4

  3. Expectation counts the mean number of successes.

The requested value is 2.5.

Checks and common pitfalls: Expectation counts the mean number of successes.

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

07 / Foundation#Your turn

X~Bin(m,1/4). Find Var(X).

m=11m=11
  • 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
List the possible values and probabilities before computing moments.
Hint 2
Use this intermediate relation.
Var(X)=mp(1−p)Var(X)=mp(1-p)
Worked solution

  1. List the possible values and probabilities before computing moments.

  2. Apply the stated relation and retain its conditions.

    Var(X)=11(1/4)(3/4)=33/16Var(X)=11(1/4)(3/4)=33/16

  3. Include the failure probability as well as the success probability.

The requested value is 2.0625.

Checks and common pitfalls: Include the failure probability as well as the success probability.

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

08 / Standard#Your turn

X~Bin(m,1/4). Find Var(X).

m=12m=12
  • 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
List the possible values and probabilities before computing moments.
Hint 2
Use this intermediate relation.
Var(X)=mp(1−p)Var(X)=mp(1-p)
Worked solution

  1. List the possible values and probabilities before computing moments.

  2. Apply the stated relation and retain its conditions.

    Var(X)=12(1/4)(3/4)=36/16Var(X)=12(1/4)(3/4)=36/16

  3. Include the failure probability as well as the success probability.

The requested value is 2.25.

Checks and common pitfalls: Include the failure probability as well as the success probability.

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

09 / Standard#Your turn

A bag has t red and 3 blue balls; select 2 without replacement. Find the probability both are blue.

t=10t=10
  • 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.
C32/Ct+32C_3^2/C_{t+3}^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    P=3/78P=3/78

  3. The unordered sample denominator counts pairs, not independent trials.

The requested value is 0.0384615384615.

Checks and common pitfalls: The unordered sample denominator counts pairs, not independent trials.

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

10 / Standard#Your turn

From t red and 3 blue balls, select 2 without replacement. Find the probability exactly one is red.

t=11t=11
  • 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.
Ct1C31/Ct+32C_t^1C_3^1/C_{t+3}^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    P=33/91P=33/91

  3. Choose one from each color; do not add a spurious ordering factor.

The requested value is 0.362637362637.

Checks and common pitfalls: Choose one from each color; do not add a spurious ordering factor.

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

11 / Standard#Your turn

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

m=4m=4
  • 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≥1−2−m1-2^{-n}\ge1-2^{-m}
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    2−n≤2−4⇒n≥42^{-n}\le2^{-4}\Rightarrow n\ge4

  3. Use the complementary event of no successes.

The requested value is 4.

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

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

12 / Transfer#Your turn

From t red and 3 blue balls, select 2 without replacement. Find the probability exactly one is red.

t=13t=13
  • 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.
Ct1C31/Ct+32C_t^1C_3^1/C_{t+3}^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    P=39/120P=39/120

  3. Choose one from each color; do not add a spurious ordering factor.

The requested value is 0.325.

Checks and common pitfalls: Choose one from each color; do not add a spurious ordering factor.

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

13 / Transfer#Your turn

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

m=6m=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≥1−2−m1-2^{-n}\ge1-2^{-m}
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    2−n≤2−6⇒n≥62^{-n}\le2^{-6}\Rightarrow n\ge6

  3. 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.

Focus on one question

End-of-lesson check and correction

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    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • 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?

    Board plan

    • Defining relation: Choose the correct model for repeated independent trials or sampling without replacement.
      P(X=k)=Cnkpk(1−p)n−kP(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.

    Anticipated thinking

    • Sampling without replacement is not generally binomial.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗