Independent, Pairwise Independent and Mutually Independent events

Question

I want to understand the difference between Independent, Pairwise Independent and Mutually Independent events. I have read multiple answers related to this like in here, here and here. Most of them talk with 3 events and explain the difference between pairwise independent and mutually independent. I understand that. But what happens when there are $n$ events?

Suppose, $A_1$, $A_2$,....$A_n$ are n events, if

$$P(A_1 \cap A_2 \cap .... A_n) = P(A_1)P(A_2)....P(A_n)$$ but they are neither pairwise independent nor mutually independent. Only the above statement holds. Now, are these events still called Independent Events? Or is there any separate nomenclature for that?

Answer 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.

Answer 2

The answer written by the OP is incorrect. The corrected statement:

Suppose it is said that some events $A,B,C,D$ are independent, it means that

$$P(A\cap B\cap C\cap D)=P(A)\cdot P(B)\cdot P(C)\cdot P(D)\\ P(A\cap B\cap C)=P(A)\cdot P(B)\cdot P(C)\\ P(A\cap B\cap D) = P(A)\cdot P(B)\cdot P(D)\\ P(A\cap C\cap D)=P(A)\cdot P(C)\cdot P(D)\\ P(B\cap C\cap D)=P(B)\cdot P(C)\cdot P(D)\\ P(A\cap B)=P(A)\cdot P(B)\\\vdots$$

Saying that it implies the first line and nothing else is very incorrect. For events to be mutually independent, that means that the probability of any intersection of any subset of the events is equal to the product of their respective probabilities. Having called a collection of events "independent" is the same as having called them mutually independent, we were just lazy and didn't include the word "mutually." There is not some third kind of independence different than pairwise and mutual independence that we refer to. Further, mutual independence directly implies pairwise independence.

Answer 3

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