Kenneth ChengMathematical modeling
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Random events and probability

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

高一必修 第二册(A版).pdf · 10.1 · PDF 235 / printed page 228

Revisit first: Statistical case study

TOPIC 01

Random events and probability

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

What you will be able to explain

  • Use random events and probability to state a justified conclusion.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Event sets

An event is a subset of the sample space; probability is its numerical measure.

Probability rules

Use complements and inclusion-exclusion with their set meanings.

P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B)=P(A)+P(B)-P(A\cap B)

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in random events and probability changes. Record a reason.

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

Binomial model: fixed n=5, independent trials, common p=0.5. E(X)=2.5, Var(X)=1.25. Bars show exact probabilities.

Binomial model: fixed n=5, independent trials, common p=0.5. E(X)=2.5, Var(X)=1.25. Bars show exact probabilities.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

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

A fair spinner has k+3 equally likely labels 1,…,k+3. Find the probability of label 1.

k=2k=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
Count favourable elementary outcomes.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P=1/5P=1/5

  3. Equal-likelihood is stated, not inferred merely from the labels.

The requested value is 0.2.

Checks and common pitfalls: Equal-likelihood is stated, not inferred merely from the labels.

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

02 / Standard#Worked example

The spinner has labels 1,…,2k. Find the probability of an even label.

k=2k=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
There are k even labels.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P=2/4=1/2P=2/4=1/2

  3. The event contains several elementary outcomes.

The requested value is 0.5.

Checks and common pitfalls: The event contains several elementary outcomes.

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

03 / Transfer#Worked example

A claim assigns probability 1+1/k to an event. Explain why it cannot be valid.

k=2k=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
Check the probability axioms before calculating.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    1+1/k>11+1/k>1

  3. All event probabilities must lie in [0,1].

It exceeds one, outside the allowed probability range.

Checks and common pitfalls: All event probabilities must lie in [0,1].

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

04 / Foundation#Your turn

A fair spinner has k+3 equally likely labels 1,…,k+3. Find the probability of label 1.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
Count favourable elementary outcomes.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P=1/6P=1/6

  3. Equal-likelihood is stated, not inferred merely from the labels.

The requested value is 0.16666667.

Checks and common pitfalls: Equal-likelihood is stated, not inferred merely from the labels.

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

05 / Foundation#Your turn

The spinner has labels 1,…,2k. Find the probability of an even label.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
There are k even labels.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P=3/6=1/2P=3/6=1/2

  3. The event contains several elementary outcomes.

The requested value is 0.5.

Checks and common pitfalls: The event contains several elementary outcomes.

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

06 / Foundation#Your turn

P(A)=k/(k+2). Find P(Aᶜ).

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
An event and its complement partition the sample space.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P(Ac)=1−3/5=2/5P(A^c)=1-3/5=2/5

  3. The complement is relative to the stated sample space.

The requested value is 0.4.

Checks and common pitfalls: The complement is relative to the stated sample space.

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

07 / Foundation#Your turn

Disjoint events have probabilities 1/(k+3) and 2/(k+3). Find their union probability.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
Disjointness removes the overlap term.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P(A∪B)=1/6+2/6=3/6P(A\cup B)=1/6+2/6=3/6

  3. Mutually exclusive does not mean independent.

The requested value is 0.5.

Checks and common pitfalls: Mutually exclusive does not mean independent.

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

08 / Standard#Your turn

P(A)=1/2,P(B)=1/3,P(A∩B)=1/(6k). Find P(A∪B).

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
Subtract the intersection once.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P(A∪B)=1/2+1/3−1/18P(A\cup B)=1/2+1/3-1/18

  3. The overlap was counted in both separate probabilities.

The requested value is 0.77777778.

Checks and common pitfalls: The overlap was counted in both separate probabilities.

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

09 / Standard#Your turn

A fair coin is tossed twice. List the event “exactly one head”.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
Use ordered outcomes because toss positions differ.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    A={HT,TH}A=\{HT,TH\}

  3. An event is a set; its probability is a number.

{HT,TH}.

Checks and common pitfalls: An event is a set; its probability is a number.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

10 / Standard#Your turn

A claim assigns probability 1+1/k to an event. Explain why it cannot be valid.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
Check the probability axioms before calculating.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    1+1/k>11+1/k>1

  3. All event probabilities must lie in [0,1].

It exceeds one, outside the allowed probability range.

Checks and common pitfalls: All event probabilities must lie in [0,1].

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

11 / Standard#Your turn

A fair spinner has k+3 equally likely labels 1,…,k+3. Find the probability of label 1.

k=4k=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
Count favourable elementary outcomes.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P=1/7P=1/7

  3. Equal-likelihood is stated, not inferred merely from the labels.

The requested value is 0.14285714.

Checks and common pitfalls: Equal-likelihood is stated, not inferred merely from the labels.

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

12 / Transfer#Your turn

The spinner has labels 1,…,2k. Find the probability of an even label.

k=4k=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
There are k even labels.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P=4/8=1/2P=4/8=1/2

  3. The event contains several elementary outcomes.

The requested value is 0.5.

Checks and common pitfalls: The event contains several elementary outcomes.

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

13 / Transfer#Your turn

P(A)=k/(k+2). Find P(Aᶜ).

k=4k=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Describe the sample space and event sets, then count or apply a probability identity.
Hint 2
An event and its complement partition the sample space.
Worked solution

  1. Describe the sample space and event sets, then count or apply a probability identity.

  2. Calculate or simplify this relation.

    P(Ac)=1−4/6=2/6P(A^c)=1-4/6=2/6

  3. The complement is relative to the stated sample space.

The requested value is 0.33333333.

Checks and common pitfalls: The complement is relative to the stated sample space.

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for random events and probability?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use random events and probability to state a justified conclusion.
    • Event sets: An event is a subset of the sample space; probability is its numerical measure.
    • Probability rules: Use complements and inclusion-exclusion with their set meanings.
      P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B)=P(A)+P(B)-P(A\cap B)
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: An event is a subset of the sample space; probability is its numerical measure.
    • Expected reasoning: Use complements and inclusion-exclusion with their set meanings.
    • Expected correction: Disjointness and independence describe different relationships.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗