1.6b Independence of Several Events

Mutual Independence

Events $A_1,\dots ,A_N\subseteq \Omega$ are said to be mutually independent if, for every single combination $S$ of the interval $[1,N]$, such that $S\subseteq \{1,2,\dots ,N\}$ and $|S|\ge 2$, the joint probability of all events $A_i$ for $i\in S$ is equal to the product of their marginal probabilities. Mathematically,
$\mathbb{P}\left[{\bigcap }_{i\in S}{A}_i\right]={\prod }_{i\in S}^{}\mathbb{P}[A_i].$

Intuition

Mutual Independence is simply an extension of the pairwise independence principle. Consider for simplification the events $A_1,A_2,A_3$.

Examples

Pairwise Independent but Not Mutually Independent

Suppose you flip a fair coin twice. Let three events be:

• A: The first flip is heads.
• B: The second flip is heads.
• C: Both flips are the same (HH or TT).

The first and the second events are independent. So are the first and the third, since $\mathbb{P}[A\cap C]=\mathbb{P}[A]\mathbb{P}[C]=\frac{1}{4}$.

However, these events are not mutually independent: if we are given that A and B occurred, the probability of C occurring is 1. These events are independent by pairs, but not mutually.