There’s no need to make assumptions about the initial numbers, because the half-life doesn’t depend on that. If you look at the three animations, those all represent the same type of nucleus. They give the same half-life, even though they start with different numbers of particles. All we’re measuring in radiometric decay is the number of decays in a given amount of time, and the decay curve — the number of decays as a function of time — is what you find experimentally. No assumptions needed!
Finding the age of a rock by isotope ratios is a little more complicated (I might write a follow-up on that), but it’s similarly straightforward. For that, you don’t even worry about how things change in time: you just assume that the entire rock formed at the same time (a reasonable assumption), and measure ratios of different types of isotopes through the entire rock. In this method, you don’t need to know how many particles there were to start with, just how many there are *now*.

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