Background
The almost-disjoint gap (ADG) is given by
$$\mathcal{D}_{\mu} \left( E_1, \ldots, E_n \right) \triangleq \mu \left( \bigcup_{k=1}^{n} E_k \right) - \sum_{k=1}^n \mu \left( E_k \right)$$
where $\mu$ is a measure and $\{ E_k \}_{k=1}^n$ are elements of a suitable $\sigma$-algebra. When the ADG equals zero we know that the events are almost everywhere pairwise disjoint.
The IE principle (for probabilities, just to skip general measure) states that
$$P\left( \bigcup_{k=1}^n E_k \right) = \underbrace{\sum_{k=1}^n P(E_k)}_{\text{Marginal Terms}} + \underbrace{\sum_{k=2}^n \left((-1)^{k-1} \sum_{\substack{I \subseteq \{1, \ldots, n \} \\ |I| = k}} P\left(\bigcap_{i \in I} E_i \right) \right)}_{\text{Joint Terms}}$$
where I have labelled what I call the marginal terms and joint terms in under braces. This tells us directly what the meaning of the ADG is: the joint terms.
$$\mathcal{D}_{P} \left( E_1, \ldots, E_n \right) = \underbrace{\sum_{k=2}^n \left((-1)^{k-1} \sum_{\substack{I \subseteq \{1, \ldots, n \} \\ |I| = k}} P\left(\bigcap_{i \in I} E_i \right) \right)}_{\text{Joint Terms}}$$
The Frechet bounds for the probability of a union given the marginal probabilities is given by
$$\max \{ P(E_k) \} \leq P\left( \bigcup_{k=1}^{n} E_k \right) \leq \min \left(1, \sum_{k=1}^n P(E_k) \right).$$
The ADG and Frechet bound together can be used to create bounds on the ADG when the marginal probabilities are given.
$$\max \{ P(E_k) \} - \sum_{k=1}^n P \left( E_k \right) \leq \mathcal{D}_{P} \left( E_1, \ldots, E_n \right) \leq \min \left(1, \sum_{k=1}^n P(E_k) \right) - \sum_{k=1}^n P \left( E_k \right)$$
Even not knowing the marginal probabilities we know the most extreme bounds for the ADG can be written in terms of the number of events.
$$1 - n \leq \mathcal{D}_{P} \left( E_1, \ldots, E_n \right) \leq 0$$
The Duck
From here is where I start making some choices to make a duck, but I don't know if what I made is actually the duck. What I want is a probability interpretation of a score on $[0,1]$ obtainable from above which says something about the amount of space or events participating in the joint terms.
Noting that $1-n \leq 0$, we can flip the bounds by multiplying by negative one:
$$0 \leq -\mathcal{D}_{P} \left( E_1, \ldots, E_n \right) \leq n-1.$$
And since $n-1$ will sometimes be greater than 1 we can divide through by $n-1$ to normalize:
$$0 \leq \frac{-\mathcal{D}_{P} \left( E_1, \ldots, E_n \right)}{n-1} \leq 1$$
Assuming I have not messed up some elementary math, which is always possible, it appears that I have a number that superficially looks like a probability.
One interpretation I am hopeful to obtain is something like $\frac{-\mathcal{D}_{P} \left( E_1, \ldots, E_n \right)}{n-1}$ being the probability of at least some overlap (i.e. intersection) among the $n$ events. If that interpretation holds, then the complement $1 + \frac{\mathcal{D}_{P} \left( E_1, \ldots, E_n \right)}{n-1}$ would be the probability of there being no overlap among the events. Both of these interpretations would be up to a set of measure zero, mind you.
Question
- Is $\frac{-\mathcal{D}_{P} \left( E_1, \ldots, E_n \right)}{n-1}$ actually a probability?
- If yes, what is its outcome space? I.e. what sort of events does it report probabilities on?
- It would be a delight for me if it were something like the probability of two or more events intersecting.
