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In the book "Introduction to Probability" by J. Charles M. Grinstead and Laurie Snell independent events are introduced in the following way: "It often happens that the knowledge that a certain event $E$ has occurred has no effect on the probability that some other event $F$ has occurred, that is, that $P (F |E) =P (F )$". This is then taken as the definition (notice that the setting in which this is done is that of discrete probabilities - if this makes any difference).

But I can't come to terms with this definition, because of the following two reasons:

a) If I have a die then the event of getting a face with one of the numbers $1,2$, if rolling once, seems intuitively to be independent of getting a face with one of the numbers $5,6$. But since these two events are disjoint, but neither is the empty space, by the above definition, they should be dependent, which seems very counter-intuitive.

b) If we chose to somehow "interpret visually" this definition, then that would mean that the sum of the probabilities of all elements in $F$ weighted with the weight $1$ is equal to the sum of all probabilities in $F\cap E$ weighted with $\frac{1}{P(E)}$, because $$\sum_{\omega\in F} \omega=P (F )=P (F |E) =\frac{P(F\cap E)}{P(E)}=\sum_{\omega\in F\cap E} \omega \cdot \frac{1}{P(E)}.$$

Or - if I use the equivalent definition of independence, $P(E\cap F)=P(E) P(F)$ - then the same interpretation of this definition would say that the sum of all elements in $F$ multiplied the sum of all elements in $E$, both weighted with the weight $1$, is again equal to the sum of all probabilities in $F\cap E$ weighted with $\frac{1}{P(E)}$.

But both these interpretations seem artificial - out of this I can't deduce, why this definition is calling $E,F$ "independent". Could you please solve this "paradox" for me ?

If I were to give a definition of independent events, I would say that $E,F$ are independent (which, for me, would mean "they don't have anything to do with each other"), if $E\cap F =\emptyset$. Why would this be a bad definition ?

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    @Jyrki: Those comments together look like an answer to me.2012-03-22
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    Re the title of your question: A big part of the goal of learning is to develop a better intuition. This is way more important than learning mere facts. I dare say that with a bit of study, the definition of independence becomes intuitively obvious.2012-03-22
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    @Jyrki: Like joriki said. +1 in advance.2012-03-22
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    For some reason that I don't quite understand, confusing "independent" and "mutually exclusive" is one of the most common mistakes made by beginners in probability theory.2012-03-22
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    @JyrkiLahtonen Ok, I understand now a). But the interpretation, as done in b), still seems to account for the "odd-ness" of the definition.2012-03-22
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    Moved the comments to an answer. Feels a bit soft to qualify as an answer, but when the aim is to build intuition, I guess this is what happens.2012-03-22
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    Essentially the [same question](http://stats.stackexchange.com/q/24877/6633) was discussed over in stats.SE just yesterday. The OP may find it helpful to read the answers there as well.2012-03-22
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    I'm with Robert, this really boggles my mind. When an *independent* team of scientists performs a study, it doesn't mean they intend to do the exact opposite experiment or come to the exact opposite results of another. When we become *independent* of our parents, we don't make decisions based on "what *wouldn't* my parents do?" When one registers as *"independent"* in the US, it doesn't mean one intends to vote the exact opposite of Republicans and Democrats (whatever that would mean). In short, do people not really understand what "dependency" means?2012-03-22
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    @RobertIsrael: I try to address the issues that confuse people in my post, if you've forgotten what they are.2012-03-22
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    If two events are disjoint, then they are very dependent to each other. If one of them happens, then the other cannot happen! So I definitely wouldn't say "they don't have anything to do with each other".2012-03-22
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    @anon But in order to be independent, the two teams of scientists shouldn't have any members in common, right? (I don't think this particular appeal to the everyday meaning of the word does the clarifying job you want it to do.)2012-03-22
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    @StevenTaschuk: the common concept between technical and non-technical uses of 'independent' is "not influenced from outside". In common use, independent scientists are not influenced by other teams. In probability, independent events are not influenced by knowing the outcome of one of them. The emphasis is on influence, not on disjointedness.2014-05-19

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