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Chulent: we first need to mention that there are two different things referred to as “kavua”. This is partially because siman 110 deals with two different types of “Rov”.
There is a rov where we have a piece which comes either from issur or heter and there is a rov of one of them. The example is 9 stores selling kosher and one store selling treif.
In this case we say “kol dparish, meruba parish”, meaning that we assume that the piece in question came from the majority. However, we also say that if it was “kavua”, here defined as that the uncertainty was known while the piece was in its original place (the store- example, you were spacing out and didn’t know which store you were in), then we do not assume the piece came from the majority, rather it is an equal safek. This is min hatora and learned from “v’arav lo v’kam” (BK 44b)
There is another rov which is the concept that if a piece is lost in a mixture, it takes on the status of the majority. This is min hatora and learned from “acharei rabim l’hatos”. Hence, if a piece of treif were to be mixed among two pieces of kosher meat, they are all permited. (We throw out one, 109:1.)
However, the rabanan decreed that certain significant items are not batul in a majority. http://www.theyeshivaworld.com/coffeeroom/topic/eggs-davar-shebiminyan
They called this “kavua”. It is sometimes referred to as “kavua derabanan”.
The case you cited with the seven cows, the maharit said that the cows which were taken to the store are ok because of the first rov we discussed. The cows in the house were not “parish” so that cannot apply. We still need to consider the second rov, to say chad btrei batul. We are not able to say that either because the pieces were significant.
So, in your house, if pieces of issur are mixed among heter, it will not be “kavua d’oraisa”, but will depend on a myriad of other factors. Basically, the whole siman 109 and much of 110.