Does Pr(Y | X=x) equal Pr(Y | do(X=x)) in a randomized experiment?

Question

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?

Answer 1

$P(Y|do(X=x))=P(Y|X=x)$ any time there are no backdoor paths between $X$ and $Y$. In a randomized trial, this is true. In an observational study, this is not (i.e., because treatment and outcome often have common causes). So you cannot simply compare $P(Y|X=x)$ and $P(Y|X=x')$ in an observational study and claim the difference is a causal effect, but you can do so in a randomized trial. In an observational study, there are a variety of ways to parametrically or nonparametrically identify the causal effect by removing or accounting for backdoor paths.

Answer 2

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?

No, exactly because experimental and observational data are very different.

Indeed in general $P(Y|do(X=x)) \neq P(Y|X=x)$ and this formalize the main difference.

The topic is vast. Shortly, considering that $Y$ and $X$ are the only variables involved in the system, only if $X$ is randomly assigned the equivalence hold.

I don't think $Pr (Y |do(X = x))$ equals $Pr (Y |X = x)$ in this [just defined] case. Why is it easy to see?

You can consider this rule like a (simplified) definition of causation.

However, in more general system, sometimes is possible to find a set of controls $Z$ so that $P(Y|do(X=x)) = P(Y|Z=z, X=x)$. This is a key rule for causal inference in observational studies. Moreover, under certain assumptions $Z$ can also be and empty set.

For more details read here:

conditional and interventional expectation

Under which assumptions a regression can be interpreted causally?