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Usually its stated that two events are independent if and only if their joint probability equals the product of their probabilities. i.e:

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

However, I was not sure if that was just a definition or if it had a proof. Usually the way I have seen it made sense is relating it to conditional independence:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

If independent then the distribution doesn't change if we are given that B has occurred:

$$P(A) = \frac{P(A \cap B)}{P(B)}$$

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

And then there is a proof for the statement but my concern is, if one of the two events has zero probability of occurring, then I was not sure what happened. For example, is the definition of independence only valid when P(A) and P(B) are non-zero? (since conditional probabilities don't really exists if the denominator is zero) Or Maybe $P(A \cap B) = P(A)P(B)$ is always true? Basically when does $$P(A \cap B) = P(A)P(B)$$ hold? Always?

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You may put it this way: two disjoint events are independent iff the probabilty of one event equals zero.

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