 A conditional probability such as \(\PP(Y = y \mid X = x)\) 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 writes \(\PP(Y = y \mid \do(X = x))\) to characterize this probability. 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. The value of the variable is determined completely by our intervention, the causal influence of the other variables being completely overridden. Graphically, we can represent the effect of this intervention by eliminating the arrows directed into the variables intervened upon. Such an intervention is sometimes described as ‘breaking’ those arrows. 