# What would be an ignorance prior of AB, given the probabilities of A and B?

Let us have two events, $$A$$ and $$B$$ whose probabilities are $$P(A)$$ and $$P(B)$$. In the absence of any other information, what would be a reasonable probability to assign to $$AB$$, that is, $$A$$ and $$B$$ being true simultaneously?

The first thing that inevitably comes to mind is $$P(AB):= P(A)P(B)$$. That is, if we have no reason to assume any of the many possible ways in which $$A$$ and $$B$$ are dependent, let us assume that they are independent. However, an "independent" seems to stand out as very special among all the possible "dependent" s. Since $$0 \leq P(AB) \leq \min(P(A), P(B))$$ it seems somehow odd, that one point in this interval would be the "default" somehow.

In which situation would we use one intersection prior over another?

If we assume that $$A$$ and $$B$$ are independent then the probability that they occur at the same time would be the same as the probability of their intersection. $$P(A\cap B)=P(A)P(B)$$. This scenario could be represented as rolling two dice at the same time. And if we further assume that $$P(A)\neq \{0,1\},\, P(B)\neq \{0,1\}$$ then $$0.