# Simple Question On Independent Events

I'm reading DeGroot and it says:

A and B are independent if and only if Pr(A|B) = Pr(A) and Pr(B|A) = Pr(B).

My question is:

• Do both equations need to hold for the two events to be independent or does only one need to hold?
• If both equations need to hold, can someone provide an example where only one holds and the other is violated?

Two events $$A$$ and $$B$$ are independent (noted $$A \perp B$$) if $$P( A \cap B) = P(A)P(B)$$, that's the definition of independence for two events.
The definition of $$P(A \mid B)$$ is $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$
\begin{align*} A \perp B &\iff P(A \cap B) = P(A)P(B) \\ &\iff \frac{P(A \cap B)}{P(B)} = \frac{P(A) P(B)}{P(B)} = P(A) \\ &\iff \frac{P(A \cap B)}{P(A)} = \frac{P(A) P(B)}{P(A)} = P(B) \\ &\iff P(A \mid B) = P(A) \\ &\iff P(B \mid A) = P(B) \end{align*}
Thus if $$P(A \mid B) = P(A)$$ then $$A$$ and $$B$$ are independent and you will also have $$P(B \mid A) = P(B)$$.