PRA: Rare event approximation with $P(A\cup B \cup \neg C)$?

The rare event approximation for event $A$ and $B$ means the upper-bound approximation $P(A\cup B)=P(A)+P(B)-P(A\cap B)\leq P(A)+P(B)$. Now by inclusion-exclusion-principle $$P(A\cup B\cup \neg C)= P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap \neg C) -P(B\cap\neg C) +P(A\cap B\cap \neg C) \leq P(A)+P(B)+P(C)+P(A\cap B\cap \neg C)$$

and now by Wikipedia, this is the general form

so does the rare-event-approximation means removal of minus terms in the general form of inclusion-exclusion principle aka the below?

$$\mathbb P\left(\cup_{i=1}^{n}A_i\right)\leq \sum_{I\subset\{1,3,...,2h-1\}; |I|=k}\mathbb P(A_I)$$

where $2h-1$ is the last odd term in $\{1,2,3,...,n\}$.

Example

For example in the case of three events, is the below true rare-event approximation?

$$P(A\cup B\cup \neg C) \leq P(A)+P(B)+P(\neg C)+P(A\cap B\cap \neg C)$$

P.s. I am studying probability-risk-assessment course, Mat-2.3117.

-

Do you mean the rare-event approximation for $P(A\cup B\cup C)$ with the three events is not unique i.e. I can approximate the original expressions in multiple ways by removing between one and all minus terms? –  hhh Apr 4 '13 at 0:17
There is $\Pr(A)+\Pr(B)+\Pr(C)$, and of course the full absolutely correct three-term expression, which is not really an approximation. Things get more interesting at, say, $5$ terms, where we can stop after $\sum \Pr(A_i)$, or $\sum \Pr(A_i\cap A_j\cap A_k)$ (or at the end, but then we are not really approximating). Note that your approximation around the third line is unduly pessimistic, you don't want to add the $\Pr(A\cap B\cap C)$ term. –  André Nicolas Apr 4 '13 at 0:22
The least upper bound is the exact answer. We want an easy to compute upper bound. In surprisingly many cases, $\sum \Pr(A_i)$ is good enough. When it isn't, $\sum \Pr(A_i)-\sum\Pr(A_i\cap A_j)+\sum\Pr(A_i\cap A_j\cap A_k)$ is often good enough. One may even have to go two steps further. –  André Nicolas Apr 4 '13 at 0:40
That's another kind of rare event approximation. It comes up when one is doing the theory of the relationship between the exponential, the Poisson, and the binomial when $np$ is "small." We end up assuming that in a small amount of time $\Delta t$, the probability of two or more events is negligible. –  André Nicolas Apr 4 '13 at 1:09