Probability of intersection involving a continuum

I’m trying to derive pascal’s rule of succession that I saw on a 3b1b video in a rigorous-like way(i don’t have a solid background in measure theory so I can’t pretend to be rigorous).
let $ ω=[0,1] × E$, $D$ a subset of $ω$, $B = [0,x]×E$ for some $x$ in $[0,1]$ and finally $S(t)={t}×E$ and $F(t)=[0,t]×E$. then I want to know if:

$P(B∩D)=int_{0}^{x}P(D|S(t))×frac{dP(F(t))}{dt}dt$

or in much looser terms assuming possible outcomes are of the form $(p,d)$ where $p in [0,1]$ and $d in Data$

$P(Data∩pin [0,x])=int_{0}^{x}P(Data|p=t)×f(t)dt$

where p is the component from the interval (the true probability of say one outcome of a coin flip in the video) and f the initial probability density function for p.
if that is right any kind of confirmation/proof is much appreciated. of it’s wrong I would love to learn the right way to deal with these kinds of probabilities that involve some continuous parameters so to speak. and if it makes sense how to use integrals to evaluate them.