Intuition behind independent events in introductory statistics

Question

I'm learning through an introductory statistics textbook, and I just can't get the intuition when the textbook mentioned this for independent events $A$ and $B$:

$\bullet$ P(A|B) = P(A)

$\bullet$ P(B|A) = P(B)

$\bullet$ P(A AND B) = P(A)P(B)

Answer 1

The usual definition of two events being independent is that

$$P(A \cap B) = P(A)P(B)$$

The conditional probability formula is

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

(with a similar form for $P(B|A)$, just $P(A)$ on the bottom instead). The first two bullets will immediately follow. An example: if $A,B$ are independent,

$$P(A|B) = \frac{P(A \cap B)}{P(B)} \overset{ind.} = \frac{P(A)P(B)}{P(B)} = P(A)$$

(where $\overset{ind.} =$ denotes where we invoke the definition of independence as in the first lines of this post).