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#1




About the independent events
Hello guys, I do not know how to interpret this.
By definition, events A and B are independent if P(A) * P(B) = P(A and B) so in a fair dice, if A = {1,2,3}, B= {1,6} then P(A) * P(B) = 1/3 * 1/2 = 1/6 = P{1} thus A and B are independent but what does "independent" mean here? why do we define independent this way? I don't understand the logic behind this. Thank you! 
#2




Intuitively, independence means that knowing whether or not one event occurred doesn't affect your knowledge of whether or not the other one occurred. Here, P[A] = 1/2. If I told you that B occurred, then the roll is a 1 or a 6, one of which is in {A}, so P[A  B] = 1/2 = P[A]. That is, being told that B occurred doesn't change your probability of A having occurred. Likewise, P[b] = 1/3 = P[B  A].
As to why we define it as P[AB] = P[A] * P[b], that approach turns out to be mathematically cleaner and more concise, as having that factorization gives us a single thing to check that has nice implications for all the components we care about. 
#3




Independent events usually overlap because otherwise the intersection is empty and has probability zero so that would require one of the events to also have probability zero.
It’s defined this way because when neither event has probability zero it’s equivalent to saying that the probability of one event does not change if you condition on the other (ie. the certainty of one of the events does not provide any more information about the probability of the other occurring). To see this just divide each side by the probability of one of the events and the other side of the equation is the conditional probability. 
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Thanks a lot! 
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