关于事件独立性定义及逆命题是否成立的技术问询

Hey there, let's break this confusion down clearly for you and your brother!

First, let's nail down the standard definition in introductory probability theory:

Two events $A$ and $B$ are defined as independent if and only if $P(A \cap B) = P(A)P(B)$

This means the equality isn't just a property that independent events have—it's the exact definition of what it means for two events to be independent. So the "converse" you're asking about is actually baked into the definition itself: if that equation holds, then by definition, $A$ and $B$ are independent events.

Now, why might you remember your undergrad professor saying the converse isn't true in general? It's almost certainly a mix-up with a related but distinct scenario:

• Maybe it was about mutual independence of three or more events. For three events, pairwise satisfying $P(X∩Y)=P(X)P(Y)$ doesn't guarantee full mutual independence (there's an additional condition required for that). That's a common point of confusion, but it doesn't apply to just two events.
• Alternatively, your professor might have been discussing edge cases in non-standard probability frameworks, but for the introductory material your brother is working through, his textbook's definition is the universally accepted standard.

To wrap it up: Your brother's textbook is totally correct. For two events, the equality $P(A \cap B) = P(A)P(B)$ is both the definition of independence and its only necessary and sufficient condition—there's no scenario where the equality holds but the events aren't independent.

备注：内容来源于stack exchange，提问作者Paul Ash