Independent, Pairwise Independent and Mutually Independent events

Solution 1:

I got the below answer.

Mutual Independence and Pairwise Independence can be defined on a collection of events only. When it is said that a collection of events is independent, it means that all the events in the collection are mutually independent.

Suppose it is said that some events $A$, $B$, $C$, $D$ are independent, it means that $$ P(A \cap B \cap C \cap D) = P(A) * P(B) * P(C) * P(D)$$ and nothing else. We cannot assume that these events are pairwise or mutually independent.

Solution 2:

We will take the following definitions

Suppose $A_1,A_2,\ldots,A_n$ are $n$ events.

Definition 1: They are pairwise independet if $$P(A_i\cap A_j)=P(A_i)P(A_j)\; \forall 1\leq i,j\leq n\;,i\not=j$$

Definition 2: They are mutually independet if $$P(A_{i_1}\cap A_{i_2}\cap\ldots\cap A_{i_m})=P(A_{i_1})P(A_{i_2})\ldots P(A_{i_m})$$ $\forall 1\leq i_1<i_2<\ldots<i_m\leq n$ , $\forall m=2,3\ldots,n$, that is, for any combination of events you choose, satisfy the product rule.

Definition 3 They are independent if $$P(A_1\cap A_2\cap \ldots \cap A_n)=P(A_1)P(A_2)\ldots P(A_n)$$

Remark :

1. Pairwise independent doesn't imply mutually independent but mutually implies pairwise.

2. Mutually independent implies independent, but not the converse because it is possible to create a three-event example in which

$${\displaystyle \mathrm {P} (A\cap B\cap C)=\mathrm {P} (A)\mathrm {P} (B)\mathrm {P} (C)}$$ and yet no two of the three events are pairwise independent (and hence the set of events are not mutually independent) George, Glyn, "Testing for the independence of three events