If we let the outcome Y be “Judy’s death” (with Y = 0 if Judy lives and Y = 1 if Judy dies) and the treatment X be “Joe’s blocking the fire escape” (with X = 0 if he does not block it and X = 1 if he does), then we are instructed to ask the following question:
Given that we know the fire escape was blocked (X = 1) and Judy died (Y = 1), what is the probability that Judy would have lived (Y = 0) if X had been 0?
Symbolically, the probability we want to evaluate is P(Y$$_{X = 0}$$ = 0 | X = 1, Y = 1). 

University of Michigan - Ann Arbor

The Model Penal Code expresses the “but-for” test as follows: “Conduct is the cause of a result when: (a) it is an antecedent but for which the result in question would not have occurred.” If the defendant fired a gun and the bullet struck and killed the victim, the firing of the gun is a but-for, or necessary, cause of the death, since the victim would be alive if not for the firing. But-for causes can also be indirect. If Joe blocks a building’s fire exit with furniture, and Judy dies in a fire after she could not reach the exit, then Joe is legally responsible for her death even though he did not light the fire.
(Symbolic representation see the child node)

Counterfactuals and the law

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

Symbolic representation of the "but-for" causes

PN is the probability that Y would be y′ had X been x′, given that, in reality, Y is actually y and X is actually x.
$$PN(x, y) = P(Y_x′ = y′|X = x, Y = y)$$
PS is the probability that Y would be y had X been x, given that, in imagination, Y is y’ and X is x’.
$$PS(x, y) = P(Y_x = y|X = x’, Y = y’)$$


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