TITLE: What is the intuition behind $\mathbb{P} (A \text{ and }B) = \mathbb{P}(A) · \mathbb{P}(B)$ if they are independent events?
QUESTION [10 upvotes]: I am unable to understand this formula intuitively.

REPLY [2 votes]: The Theory of Probability is an axiomatic structure. As you can see in Definition 1 in section 5 in Kolmogorov's Theory of Probability, this formula defines what we mean by independent events. That is, it is not a theorem and it is not based on anything empirical. It just is. A definition is not something one arrives at. It is something one takes as given.
Saying "N events are independent" and "the probability of all of them happening is the multiplication of the probabilities of each" mean the same thing.