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A simple example is the two definitions for independence of events:

1. A and B are independent iff $P(A\cap B) = P(A)P(B)P(A)P(B)$
2. A is independent from B iff P(A|B$P(A\mid B) = P(A)P(A)$

Some presentations start with Definition 1, which is entirely uninformative: nothing in it explains why on earth we bother discussing this. In contrast, Definition 2 says exactly what "independent" means: knowing that B has occured does not change the probability that A occurs as well.

A reasonable introduction to the subject should start with Definition 2; then observe there is an issue when P(B)=0, and resolve it; then observe independence is symmetric; then derive Definition 1.