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When I check for definition of independent events I get the following definitions

Type 1 (Reference )

When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring.

Type 2

$P(A \cap B) = P(A)P(B)$

Then it follows some useful information such as ( Reference page 14)

a) Do not say that two events are independent if one has no influence on the other
b) No circumstances say that A and B are independent if $ A \cap B = \phi $ (this is the statement that A and B are disjoint, which is quite a different thing!)
b) Also, do not ever say that P(A ∩ B) = P(A) · P(B) unless you have some good reason for assuming that A and B are independent (either because this is given in the question, or as in the next-but-one paragraph).

Example
Consider example where events are not independent

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My conclusion is that $P(A/B) \ne P(A)$ and $P(B/A) \ne P(B)$ . It says given precondition of event B has an effect on P(A) ($P(A/B) \ne P(A)$) and given precondition of event A has an effect on P(B) ($P(B/A) \ne P(B)$) . So these are dependent events

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My conclusion is that $P(A/C) = P(A)$ and $P(C/A) = P(C)$ . It says given the precondition of an event C has no effect on P(A) ($P(A/C) = P(A)$) and an event A has no effect on P(C) ($P(C/A) = P(C)$). So these independent events


Doubt
1) Mathematically we can say whether two events are independent by formulas such as $P(A/C) = P(A)$ and $P(C/A) = P(C)$ or $P(A\cap C) = P(A)P(C)$ . But is it possible to show meaning using Venn diagrams as we show $P(A \cup B )$ etc.? Can we make some statements about the participating event sets if know prior two events are independent. I am looking for getting a conceptual meaning derived from the formula with visual identification from Venn diagram if possible


2) But if you check the definitions I got from two sources as mentioned above as type 1 and type 2, type 1 says

When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring.

and type 2 says

Do not say that two events are independent if one has no influence on the other.No circumstances say that A and B are independent if $ A \cap B = \phi $ (this is the statement that A and B are disjoint, which is quite a different thing!)


What is the difference between influence of events on other and affecting other event as mentioned above? What is meaning of two independent events which never affects each other but influence each other? I am looking for a precise conceptual definition or identification method for independent events

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marked as duplicate by BCLC, José Carlos Santos, Chris Custer, cansomeonehelpmeout, Leucippus May 27 '18 at 0:41

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I think that

"Do not say that two events are independent if one has no influence on the other"

is just an advice that should be followed if it is asked on e.g. some exam about probability theory: "are $A$ and $B$ independent?".

If they are indeed independent then your answer must be: "yes, because $P(A\cap B)=P(A)\times P(B)$,..." eventually followed by some argumentation.

Further in real life there is a difference between "influence of event $A$ on event $B$" and "influence of the probability of event $A$ to occur on the probability of event $B$ to occur".

It might happen that first mentioned is there, while the second mentioned lacks.

In probability theory we have modeled reality and the first mentioned has no place in our model. Two events are by definition independent if the (stronger) second mentioned influence lacks.

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