id: 06604802
dt: j
an: 2016e.00910
au: Kachoyan, Bernard
ti: If you don’t find anything, is there really nothing there?
so: Parabola 52, No. 1, 6 p., electronic only (2016).
py: 2016
pu: AMT Publishing, Australian Mathematics Trust, University of Canberra,
Canberra; School of Mathematics \& Statistics, University of New South
Wales, Sydney
la: EN
cc: K60 K70
ut: stochastics; probability theory; conditional probability; Bayes’ theorem;
a priori; a posteriori; probability distributions; Katz distribution;
uncertainty; real-world problems
ci:
li: https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-52-2016/issue-1/vol52_no1_1.pdf
ab: From the text: Let us suppose that you are looking for some objects in a
particular area. Now also suppose that you find nothing after searching
the area. Is there really nothing there? Or, more generally, if you
find $N$ objects, then how many objects are there left to be found?
Equivalently, how many were actually there to be found in the first
place? The objects might be physical objects such as people in a search
and rescue operation, or animals, schools of fish, ore pockets or oil
deposits. Or they might be “virtual" things like bugs in computer
code, faults in manufactured products, etc. If the sensors you use,
whether some electronic means or even your own eyes, are perfectly
capable and will find an object with 100 \% probability, then the
answer is clearly trivial. You have found all that were there to be
found. But what happens if your sensors are less than perfect, either
inherently so, or perhaps because the objects may be concealed or the
environment difficult to search? Moreover, if you know before the
search that there are $M$ objects there and you find $N$, then you also
know that there are $M-N$ left after the search. But what if the number
of present objects is uncertain and can be expressed as a probability
distribution on the number of objects thought to be present.
Intuitively, if we find $N$ of these, then there should be a resultant
distribution of the number of objects left. We will see how these two
distributions can be connected, together with the probability of
finding each object, through what is known as Bayes’ Theorem.
rv: