In order to show that H causes e you would have to show that the probabilities always factor as P(e & H) = P(H)P(e|H) and not as P(e & H) = P(e)P(H|e).
Both of these are mathematical identities. It is not possible for one to hold and not the other; both are always true.
Both of these are mathematical identities. It is not possible for one to hold and not the other; both are always true.
Causal analysis of probabilities is a lot more complicated.