Consider the following problem:
You have two coins each with it's own weight (probability of giving heads). Somebody will flip the coins for you in another room (you trust them). You can either ask them to flip both coins and they will tell you if they are both heads or not both heads. Or you can ask them to flip only the first coin (you can't only flip the second) and they will tell you if it was heads or not.
How should you proceed to find an estimator of the weight of the second coin. You are allowed to ask the person multiple times. The main goal is to make the estimator not depend on the weight of the first coin. Is this possible?
An aside for motivation of this problem: This is analogous to how measurements in quantum mechanics with linear loss are done. The loss is the first coin, the measurement is the second coin. We can never get rid of the loss but we can do a trivial second measurement (gives always heads) and then the real measurement. Particularly, this is the problem of measuring the polarization degree of freedom of photons from a heralded single photon SPDC source.