Kenneth ChengMathematical modeling
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Each of three independent games is won with probability 1/4. Find the probability of at least one win.

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TOPIC 01

2022 JM01

Adding three win probabilities double-counts outcomes with multiple wins.

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01 / Standard#Your turn

Each of three independent games is won with probability 1/4. Find the probability of at least one win.

Official paper · jm01-2022 · I.8 · PDF 2

Official original and suggested answers ↗ · Suggested answer PDF page 5

Skills and prerequisite lessons
  1. Option A1/641/64
  2. Option B27/6427/64
  3. Option C35/6435/64
  4. Option D37/6437/64
  5. Option E43/6443/64

Working and explanation

BUILD THE REASONING

Hint 1
Use the complement of losing all games.
Hint 2
Each loss has probability 3/4.
Worked solution

  1. Multiply the three loss probabilities.

    P(no wins)=(3/4)3=27/64P(\text{no wins})=(3/4)^3=27/64

  2. Subtract from one.

    P(at least one)=1−27/64=37/64P(\text{at least one})=1-27/64=37/64

D: 37/64.

Checks and common pitfalls: Adding three win probabilities double-counts outcomes with multiple wins.

Reasoning checklist · self / teacher assessment
  • Teaching assessment checklist, independently authored. Use the original paper for official marks.
  • Multiply the three loss probabilities.
    P(no wins)=(3/4)3=27/64P(\text{no wins})=(3/4)^3=27/64
  • Subtract from one.
    P(at least one)=1−27/64=37/64P(\text{at least one})=1-27/64=37/64

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