Probability of Human Error Happening!

Problem Statement: There are 8 boxes full of balls of different colors, out of which only one contains a red ball. A person starts searching for that red ball by opening boxes one by one. What is the probability that person searched all boxes but couldn't find the red ball because of human error. i.e. he overlooked the box containing red ball n missed it.

How can I find the precision or accuracy of that person considering probability of human error?


Smelly poop man with doo doo pants.
two important points:

a) as an abstract problem, we're still missing some information here if you want an exact numeric answer. namely, how many balls are in each box and how many different colours are there. kind of like "there are 10 balls in each of 8 boxes. 3 of them are blue, 6 of them are green..." or something like that. we could make the probability as a function of n number of balls and c different colours though to get a general solution though. we'd still need to know if, after looking at a ball, the person tosses it out or puts it back in the urn since that would change the probability rates. we just need more info to solve the problem.

in general, the solution will look like: P(not finding the red ball) = 1 - P(finding the red ball)

b) (**the most important part**) probability is an abstract measure of chance with no inherent meaning. whether you want to call it "human error" or something else is a matter of research design and not mathematics (i.e. you'd need to rule out alternative explanations that would attribute lack of accuracy to human error). for instance, in a very dumb example: if in your research design the box that has the red ball has a larger number of balls of other colours then there is a greater chance that he/she will miss it. in this case, the lack of accuracy is a function of both the number of balls AND human error AND more stuff. now you can fix that easily by making all the number of balls the same... or probably not. it depends on how you operationally define "human error". conclusion i guess would be this is an exercise in the logic behind your research design rather than in just math.
The color of balls is not important. The person was searching for something(in this case red ball) and he couldn't find that. What is the probability that he searched for "something" in all 8 boxes but couldn't get it provided that only one of the boxes contained that "something"?


Smelly poop man with doo doo pants.
i (or anyone else) still needs two pieces of information to complete this problem with an exact numeric solution. first, i need to know how many balls are in each box that are not red. second, i need to know whether you assume the person looks at whatever ball he/she looked at and puts it back inside the box or whether he or she just tossed it out and kept on looking. that would tell me (or anyone else) if this solution is binomial or hypergeometric and, hence, i can calculate its exact probability.
you may want to look at the "urn ball" problem, which is a generic term in statistics/probability to refer to the kind of question you're asking.

edit: this is actually a little bit more complicated than i initially thought because by the way you phrased it ("the probability that he searched for "something" in all 8 boxes but couldn't get it provided that only one of the boxes contained that "something") implies that these are now conidtional probabilities.... probability of "missing something" or "overlooking something" given that (conditioned on) there is "something in only one box", correct? yeap, to get the probability you want we'll need to use bayes' theorem.
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