Probability P(A) = n(A)/n(U) (when all outcomes are equally likely)
Addition Theorem・Complementary Event
P(A∪B) = P(A) + P(B) − P(A∩B)
If mutually exclusive, P(A∪B) = P(A) + P(B)
Complementary event: P(Ā) = 1 − P(A)
Example①
Probability of rolling a multiple of 2 or a multiple of 3 on a die:
P(multiple of 2)=3/6、P(multiple of 3)=2/6、P(6)=1/6 → 3/6+2/6−1/6 = 2/3
Probability of rolling a multiple of 2 or a multiple of 3 on a die:
P(multiple of 2)=3/6、P(multiple of 3)=2/6、P(6)=1/6 → 3/6+2/6−1/6 = 2/3
Example② (Complementary Event)
Probability of getting at least one head with 2 coins: 1 − P(both tails) = 1 − 1/4 = 3/4
Probability of getting at least one head with 2 coins: 1 − P(both tails) = 1 − 1/4 = 3/4
Key Points
- “At least” → complementary event
- 0≤P(A)≤1、P(U)=1、P(φ)=0
Practice Problems
- Choose one number from 1〜10. What is the probability of choosing a multiple of 3?
- Choose one number from 1〜20. What is the probability of choosing a multiple of 4 or 6?
- Roll a die twice. What is the probability that the product is even? (Use the complementary event.)
Answers
- 3/10
- 5 multiples of 4 + 3 multiples of 6 − 1 multiple of 12 = 7 → 7/20
- 1−P(both odd)=1−(1/2)²=3/4