In Halachah there's a tremendously powerful concept of Bitul BeRov (nullification among the majority). It goes like this: If you have three identical pieces of meat, and you know for a fact that one is not Kosher but it's not known which one is the non-Kosher one, you can eat all of them because of Rov: You can eat the first because Rov dictates that one of the two you're leaving on the table is probably the non-Kosher one; you can eat the second because the one still on the table or the one you already ate is probably the non-Kosher one; and you can eat the third one because you probably already ate the non-Kosher one. It's generally accepted that you must eat them one at a time for this to work, but the Rosh (Hulin 7:37) is of the opinion that, in fact, you can eat them all together, because once Bitul takes effect, the Isur (prohibited item) becomes Heter (a permitted item).
How do we apply this today? Is it straight textbook? Are there caveats added today? If it's textbook, which opinion do we follow? Finally, how far do we carry this - does it carry to other areas of Halachah, or do we restrict it to the classic case?
