The Book of Why: The New Science of Cause and Effect (Penguin Science)

by Judea Pearl

Cancer can come about in two ways: in the presence of Tar or in the absence of Tar. If we force a person to smoke, then the probabilities of these two states are P(tar | do(smoking)) and P(no tar | do(smoking)), respectively. If a Tar state evolves, the likelihood of causing Cancer is P(cancer | do(tar)). If, on the other hand, a No-Tar state evolves, then it would result in a Cancer likelihood of P(cancer | do(no tar)). We can weight the two scenarios by their respective probabilities under do(smoking) and in this way compute the total probability of cancer due to smoking. The same argument holds if we prevent a person from smoking, do(no smoking). The difference between the two gives us the average causal effect on cancer of smoking versus not smoking.