On page 435 of Cosma Shalizi's advanced data analysis book (link: https://www.stat.cmu.edu/~cshalizi/ADAfaEPoV/ADAfaEPoV.pdf), he states the following about randomized experiment for causal inference:
to examine the contrast between Pr (Y |do(X = x))$ and $Pr (Y |do(X = x')), we use an independent source of random noise to decide which experimental subjects will get do(X = x) and which will get do(X = x'). It is easy to convince yourself that this makes $Pr (Y |do(X = x))$ equal to Pr (Y |X = x).
I don't think $Pr (Y |do(X = x))$ equals $Pr (Y |X = x)$ in this case. Why is it easy to see?
Also, Shalizi says one paragraph below:
Experimental evidence is compelling, but experiments are often slow, expensive, and difficult. Moreover, experimenting on people is hard, both because there are many experiments we shouldn’t do, and because there are many experiments which would just be too hard to organize. We must therefore consider how to do causal inference from non-experimental, observational data.
If $Pr (Y |do(X = x)) = Pr (Y |X = x)$ in randomized studies, doesn't it mean that we can already estimate the causal effects by comparing $Pr (Y |X = x)$ and $Pr (Y |X = x')$, which is purely observational data, and there's no intervention needed in the first place?