# Definition of Independence in probability and how its affected if one of the events has zero probability

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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