Search & Probability
Above is a $2 \times 2$ search grid constructed to search for a lost object.
We begin by not knowing anything and hence if F = finding the object in a particular square, $P(F) = \frac{1}{4}$. Figure A.
We check $1$ square and we don't find the object (that's figure B with one square redded out). We now update $P(F)$ for the remaining $3$ squares: $P(F) = \frac{1}{3}$.
We then search a 3rd square (figure C) and update $P(F)$ accordingly, $P(F) = \frac{1}{2}$
From what I can gather, this is a brute search method and although we can compute the probabilities (vide supra), they don't allow us to refine the search (they do not aid us in any way at all because all the options are equiprobable).
Question: Is there a way we can "remedy" this i.e. (somehow) make $P(F)$ different for each square, which would allow us to shorten the path to finding the lost object (vide infra)
2 answers
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Is there a way we can "remedy" this
No, not without additional information. If you truly do not know anything about the location of the lost object other than it is not in any of the squares you searched, then there is always an equal chance of it being in each of the unsearched squares.
How about if I randomly (???) sweep sections of the squares left. Divide the 3 remaining squares into equal parts and conduct variable searches in these parts.
You are trying to get something from nothing. You have no information about the location of the object, so the only way to get information is to search. Every place you look has the same chance of having the object in it as any other place of the same size.
If you break up the 3 remaining squares into 3 equal boxes each, then you now have 9 boxes the object could be in. Each box is 3x easier to search, but also with 3x less chance the object is there.
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