 A conditional probability such as \(\Pr(Y = y \mid X = x)\) gives us the probability that Y will take the value y, given that X has been observed to take the value x. Often, however, we are interested in predicting the value of Y that will result if we intervene to set the value of X equal to some particular value x. Pearl (2009) writes \(\Pr(Y = y \mid \ido(X = x))\) to characterize this probability. The notation is misleading, since \(\ido(X = x)\) is not an event in the original probability space. It might be more accurate to write \(\Pr_{\ido(X = x)} (Y = y)\), but we will use Pearl’s notation here. What is the difference between observation and intervention? When we merely observe the value that a variable takes, we are learning about the value of the variable when it is caused in the normal way, as represented in our causal model. Information about the value of the variable will also provide us with information about its causes, and about other effects of those causes. However, when we intervene, we override the normal causal structure, forcing a variable to take a value it might not have taken if the system were left alone. Graphically, we can represent the effect of this intervention by eliminating the arrows directed into the variable intervened upon. Such an intervention is sometimes described as “breaking” those arrows. As we saw in Section  3.1, in the context of a SEM, we represent an intervention that sets X to x by replacing the equation for X with a new one specifying that \(X = x\).