What percentage of systematic data starts to be considered random There is a set of 100 different objects in a box. I pull out 99 of them randomly and place them on a table and ask a person to take a look, then I put all objects back into the box. After a short (5 minutes) interval I pull out another 99 random objects from the same box. The person is asked to say whenever an object is the same as it was before. If the missing object did not stand out from all others on the first go, then the person will answer that the objects are all the same.

How many objects should there be in the box and how many should be placed on a table, assuming there is an equal chance for every object to be pulled out, in order to trick the average person into thinking that all or almost all of the objects are different? What if the process is repeated 6 times, every 5 minutes, would it have any noticeable effect on the results?
Tests of randomness like (my favorite) Kolmogorov complexity will tell you how random those objects seem scientifically. (I'm not certain but I think your describing chaotic behavior rather than random behavior because the random objects can only be rearranged rather than replaced.)

Aside from mathematics this is an active area of psychological research. Those with disillusion illnesses like schizo typical often have the trouble identifying when a pattern is random. The sick brain seeking to bring the universe into order or create a new reality to match its broken perceptions deludes itself into believing that random events happen as per a control. This is called Pareidolia (when there is something which might vaguely resemble a pattern) and Apophenia (when the data is obviously random). 

Your test would be measuring Pareidolia, Apophenia as well as Pattern recognition.