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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?

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  • $\begingroup$ 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 $\endgroup$ Nov 13, 2020 at 1:02
  • $\begingroup$ Welcome to MathSE. This tutorial explains how to typeset mathematics on this site. $\endgroup$ Nov 13, 2020 at 8:37
  • $\begingroup$ 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 $\endgroup$
    – PwNzDust
    Nov 13, 2020 at 8:45

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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.

1/16 is equal, looking back at the sample space Ω of 4, to P(the) * P(a).

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 $|A|*|B|/Ω*Ω$?

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