# Independent, Pairwise Independent and Mutually Independent events

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?

• For a silly example, consider what happens if $A_1=\emptyset$ in which case trivially $P(A_1\cap A_2\cap\dots A_n)=P(A_1)P(A_2)\dots P(A_n)=0$, regardless of the nature of the relationships between $A_2,\dots,A_n$ – JMoravitz Aug 22 '18 at 5:40
• Sorry, I didn't get how it answers my question? Or, are you just adding an example? How does it help? – Nagabhushan S N Aug 22 '18 at 5:41
• I'm saying if you allow one of the events to be impossible, then any and every choice of $A_2,\dots,A_n$, in particular those which are not mutually nor pairwise independent, will satisfy the condition you give. They are neither pairwise nor mutually independent, and we do not have much if any information as to anything else related to how they act, so it seems unnecessary to give such a situation a name. – JMoravitz Aug 22 '18 at 5:52
• Oh! Okay. Understood. But you're giving one situation where it doesn't make sense. That doesn't mean there aren't any situations where it makes sense. The equation might hold true for some situation. Right? – Nagabhushan S N Aug 22 '18 at 6:13

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.

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 $$P(A_{i_1}\cap A_{i_2}\cap\ldots\cap A_{i_m})=P(A_{i_1})P(A_{i_2})\ldots P(A_{i_m})$$ $$\forall 1\leq i_1 , $$\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 independet, but not te reciprocal 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

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.