Scenario:
Multiple agents process orders simultaneously. Orders are not shared between agents. Each agent is assigned a sampling percentage. Let say agent A has sampling percentage of 10% and B has 20%. Both process 100 orders each. We want to audit around 10 orders from A and 20 from B.
Orders are received in a continuous flow. It is important to make QC or not to QC decision right away (as soon as agent finishes his/her work). If an order is to be audited it moves to QC queue.
Problem:
How do we ensure that order are audited as per sampling percentage assigned to each agent? We want to randomize. It is important. For example, one solution is to audit every 10th order from agent A and every 5th order from agent B. However, that makes auditing process predictable. Agent A can focus more on every 10th order and achieve higher accuracy.
Another solution is to create groups of 100/p where p is sampling percentage. Then select one randomly. However, we do not want to go with this option if alternatives are available.
Is there any other sampling theory that can help?
You could make an independent random choice for each order of whether to audit it, according to the agent's sampling percentage. (For example, audit each of agent A's orders with probability $0.1$). Over time, the actual sampling percentages will approach the desired sampling percentages. This convergence is guaranteed in the sense that it happens with probability 1.