Simple Question On Independent Events

Question

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?

Answer 1

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

Then

\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 $P(B \mid A) = P(B)$.
To answer your first question : if one holds the other one holds too.

This can be found on Wikipedia here.