Independence: P(A∩B) = P(A)P(B) holds ⟺ P(A|B) = P(A)
Example ①
P(A)=1/2、P(B)=1/3、and A and B are independent. Find P(A∪B).
P(A∩B)=1/6 → P(A∪B)=1/2+1/3−1/6=2/3
P(A)=1/2、P(B)=1/3、and A and B are independent. Find P(A∪B).
P(A∩B)=1/6 → P(A∪B)=1/2+1/3−1/6=2/3
Key Points
- Independent ⟺ P(A∩B)=P(A)P(B)
- If A and B are independent → Ā and B are also independent, A and B̄ are also independent, and Ā and B̄ are also independent
- Mutually exclusive (P(A∩B)=0) ≠ independent (do not confuse them)
Practice Problems
- P(A)=2/3、P(B)=3/4、and they are independent. Find P(Ā∩B̄).
- P(A)=0.4、P(B)=0.3、P(A∩B)=0.12. Are A and B independent?
- A coin is tossed twice. Are the event that the first toss is heads (A) and the event that both tosses are the same (B) independent?
Answers
- P(Ā)=1/3、P(B̄)=1/4、so because they are independent, P(Ā∩B̄)=1/12
- P(A)×P(B)=0.12=P(A∩B) → independent
- P(A)=1/2、P(B)=P(HH)+P(TT)=1/2、P(A∩B)=P(HH)=1/4=P(A)×P(B) → independent