Whenever, do(𝑡) appears after the conditioning bar, it means that everything in that expression is in the post-intervention world where the intervention do(𝑡) occurs. For example, 𝔼[𝑌 | do(𝑡), 𝑍 = 𝑧] refers to the expected outcome in the subpopulation where 𝑍 = 𝑧 after the whole subpopulation has taken treatment 𝑡 . In contrast, 𝔼[𝑌 | 𝑍 = 𝑧] simply refers to the expected value in the (pre-intervention) population where individuals take whatever treatment they would normally take ( 𝑇 ). This distinction will become important when we get to counterfactuals in