Section 1.4: Independent Events
Understand when events don't affect each other's probabilities.
Definition of Independence
Two events A and B are independent if knowing one occurred doesn't change the probability of the other.
Product Definition
Most common definition
Conditional Definition
Equivalent if P(B) > 0
Intuition
If A and B are independent, learning that B occurred gives you NO information about A. The events don't "influence" each other.
Independence ≠ Mutually Exclusive
This is a common exam trap! These concepts are almost opposites:
Independent Events
- • P(A ∩ B) = P(A) × P(B)
- • Can occur together
- • Knowing one tells nothing about the other
- • Example: Two coin flips
Mutually Exclusive Events
- • P(A ∩ B) = 0
- • Cannot occur together
- • Knowing one tells you the other didn't happen
- • Example: Rolling 3 or 5 on one die
Key Fact
If A and B are mutually exclusive with P(A) > 0 and P(B) > 0, they are NEVER independent.
Proof: P(A ∩ B) = 0, but P(A) × P(B) > 0. So P(A ∩ B) ≠ P(A) × P(B).
Multiple Independent Events
For n independent events A₁, A₂, ..., Aₙ:
Example: Coin Flips
Flip a fair coin 5 times. P(all heads) = ?
Each flip is independent with P(H) = 0.5
P(HHHHH) = 0.5⁵ = 1/32 = 0.03125
Independence and Complements
If A and B are independent, then so are:
Useful Application
P(at least one of independent events) = 1 - P(none of them)
Example: P(at least one head in 5 flips) = 1 - P(no heads) = 1 - (0.5)⁵ = 31/32
Mutual vs Pairwise Independence
Pairwise Independent
Every pair is independent:
- P(A∩B) = P(A)P(B)
- P(A∩C) = P(A)P(C)
- P(B∩C) = P(B)P(C)
Mutually Independent
All subsets are independent:
- All pairwise conditions +
- P(A∩B∩C) = P(A)P(B)P(C)
Exam Note
Pairwise independence does NOT imply mutual independence! There exist examples where all pairs are independent but the triple is not.