Mathematical definition of Independence

In a world where everyone is racing to be independent, let us dig mathematically what independence really means?

Technically, independence is a relation about information. This basically means occurrence of one event provides no context or effect on occurrence of other event.

Now consider a large sample space and imagine two disjoint events A and B within it.

Since, the events are disjoint (not sharing anything), we can intuitively guess, P(A⊓B) = 0

However, probability of the events happening individually remains greater than zero,

i.e. P(A) > 0 and P(B) > 0.

What this means is if we know that event A has occurred then we surely know that we event B has not occurred, which makes the two events dependents on each other. Hence, the illusion of being disjoint in a large sample space does not hold up to being independent.

For being truly independent, the equation must be P(A⊓B) = P(A)xP(B).

A sample of this would be the probability of tossing a coin resulting in head or tails is completely independent of the fact whether it will rain tonight.

We live in an interconnected world, where our subconscious is exposes to the interaction with outside world, be it visuals, sounds, texts, etc. Hence, the very notion of being independent in our mind comes from this exposure. We quickly try to evaluate ourselves based on certain parameters. However, this very evaluation beats the purpose of independence.

To be truly independent, we must be connected with the Self, where we perceive the events of outside worlds as mere observations, which do not alter the state of Self, which is anyway eternal constant.