Suppose I have $n=1,N$ independent probability measurements of whether an event $A$ occurred, i.e. $P(A|n)$. What is the best expression for the probability that the event at all?
For example, an object goes down a conveyor belt and is imaged by a set of $N$ scanners, each of which takes an image of a unique part of the object and returns a probability that it shows a defect. For example, we have 5 scanners which return probabilities $[0.01,0.04,0.7,0.06,0.1]$ of there being a defect. To be clear, scanner 1 returns a $P(1)=0.01$ that its section has a defect, scanner 2 has $P(2)=0.04$, etc.
For now, let's assume I have no knowledge of how often a defect happens. Treat each scanner's result as independent, and a defect viewed by any scanner means there is a defect present and the object must be rejected. Rather than apply a probability threshold for each scanner that returns $P(A|n)=0$ or $1$, we want to combine all the scanner outputs to yield a final probability that this object has a defect. I believe the right way is
\begin{equation} P(A) = 1-\prod_{n=1}^N[P(A^c|n)] \end{equation}
since each scanner returns $P(A^c|n)=1-P(A|n)$ that the object is not defective, yielding $P(A)\approx 0.76$.
Yes, $$\begin{aligned}P(\textrm{Defect in some part})&=P\bigg(\bigcup_{n=1}^N\{\textrm{Part }n\textrm{ has a defect}\}\bigg)=\\ &\stackrel{\textrm{De Morgan's}}{=}1-P\bigg(\bigcap_{n=1}^N\{\textrm{Part }n\textrm{ does not have a defect}\}\bigg)= \\ &\stackrel{\textrm{Indep.}}{=}1-\prod_{n=1}^NP(\textrm{Part }n\textrm{ does not have a defect})\approx 0.7588\end{aligned}$$