# Simple probability question on product rule of independent events

I would need to clarify a point on the intuition behind the product rule related to elementary probability:

Given a sample space Ω, we have that the probability of a conjunction of events $$P(A,B) = P(A) \cdot P(B)$$ if and only if $$A$$ and $$B$$ are statistically independent.

We can measure the probability of an event $$E$$ using set theory in the following way: $$P(E) = \frac{|E|}{|\Omega|}$$

For instance, given a bag of words containing "table chair the a" and assuming they are all statistically independent from each other, $$P(\text{table,chair}) = \frac{|\text{event table}|}{4} \cdot \frac{|\text{event chair}|}{4}$$

What is the justification behind defining the probability of joint independent events as the product of the probability of the two simple events? Is there a way to formally prove the result using set theory, except deriving it from the case of the conditional probability?

• I think a good place to start building intuition is by visualizing probabilities using a Venn diagram. Check out this video: youtube.com/watch?v=pV3nZAsJxl0 Nov 13, 2020 at 1:02
• Welcome to MathSE. This tutorial explains how to typeset mathematics on this site. Nov 13, 2020 at 8:37
• Thanks for the answers and the formal corrections. I still think I have not properly figured out why $P(A,B)$ should be equal to the product of the two, I specified this in the following answer Nov 13, 2020 at 8:45

After having watched the some of the videos, I think it is somewhat more clear, but not entirely: for instance, in the example I made of "table chair the a", I now get how, to measure set theoretically $$P(the,a)$$ [the probability that from a bag of words I extract the and then, from the same words, I extract a]. I have to measure the cardinality of the new event over a new sample spaces that has as simple events all the possibile results of the extraction of two words from that bag:
The number of possibile extraction of two words will be 16. So, in a space $$Ω$$ of 16 possible outcomes, the probability of the outcome(simple event) "the-a" will be 1/16.
But how do I arrive to the general rule that tells me that, given an event A and an event B and a sample space $$Ω$$, then the probability of getting A and then B is equal to the probability of $$|A|*|B|/Ω*Ω$$?