I was recently introduced to conditional and independent probabilities, but I cannot prove the "Conditional probability formula" in terms of anything rigorous rather than with the use of examples. I have tried induction, but it does not seem to work. Since I was not been able to prove it I think that it is considered as an axiom of Statistics (If it's the case, then dismiss the question). However, even if the case, it still seems hard to me to assume this tool as an axiom.
Here is the formula: P(A ∩ B) = P(A|B)P(B).
Thanks in advance!
Edit: I was not able to write the formula for conditional probability directly so I had to copy and paste from another website. Here is how it looks: P(A|B)=P(A ∩ B)/P(B), where P(B)>0 (zero)
I don't believe there is a 'rigorous' proof, the definition of conditional probability comes from the intuition behind what you are looking for. Conditional probability, $P(A|B)$, means you are looking for the probability of a certain event ($A$), given a certain amount of information ($B$).
Any time you are looking for the probability of just event $A$ you are assuming an underlying probability space $\Omega$. Therefore, $P(A)$ can also be viewed as the $P(A|\Omega) = \frac{P(A \cap \Omega)}{P(\Omega)}$ where $P(\Omega) = 1$ and $P(A \cap \Omega) = P(A)$.
Moreover, $P(A|B)$ assumes that you are still interested in finding $P(A)$; however, your sample space $\Omega$ is now being restricted only to the event $B$.
With this is mind, the probability of interest becomes $P(A \cap B)$; that is the probability of both A and B occurring. However, you still have to divide by $P(B)$ because the underlying probability space no longer has probability 1.