system
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human
Would masses of 1kg and 0,5 kg of the same radioactive material take the same time to decay?

First of all, is it even possible for something to decay 100% via radioactive decay, since half of half of half... seems like a convergent series with the decayed mass approaching one, but never really getting there... But then again, you eventually would get to one atom, and that atom (given that it's radioactive) has to decay sometime, right? That, I guess, would take an incredibly long time, but since this is purely theoretical it doesn't really matter. My brain already hurts, but lets continue...

So the intuitive thinking would be that the 0,5 kg mass will obviously decay faster, because it has less mass, duh! But I thought about it for a bit and came up with this scenario: you have a mass of 1 kg; lets say it takes a billion years for it to fully decay. Then you split that mass into 2 pieces of 0,5 kg. It would still take a billion years to decay, nothing has really changed, has it? I don't suppose it has to do anything with mass/surface ratio, does it? So now, you have two masses of 0,5 kg which will take the same time to fully decay as the 1 kg mass, a million years. 

Now, I know that half-life is purely probabilistic, meaning that in theory, a radioactive material could retain 100% of its original mass after a billion years, even if its half-life is thousands of times shorter. That, of course, has such an astronomically small chance of happening, that it's basically impossible. So, how does radioactive decay and the whole half-life thing work when the number of atoms becomes low enough for our predicted function of mass over time starts getting significantly inaccurate? 

Because each atom has a 50% chance of decaying in the half-life of the element, it's all fine while we have ~10²⁴ atoms because the higher the sample size, the more accurate the results. However, when we get to a smaller number of atoms, the amount of mass over time starts to deviate from our perfect-50%-each-period-of-half-life assumption and basically, we're lost, right? I'm guessing that as the mass approaches the really small numbers of atoms, the course of the decay could go either way. It can go slightly faster, it can go slightly slower. But, if we did this 1 and 0,5 kg masses experiment an infinite amount of times, the average time of full decay would be exactly the same, right?

So, I obviously want the question in the title answered, but I also want someone to clear radioactive decay and the half-life thing a bit. Please correct me on anything that is incorrect in this post, cause I'm still in highschool and have a pretty basic understanding of physics, with a bit of youtube videos and such on top. I tried to get some of this answered by googling but its all so basic and doesn't really get to explaining my question. Also, I didn't really try wikipedia because shits like me with a highschool understanding of physics can and do write all sorts of unimaginably incorrect things on topics they once read half and article about. Thanks.
gpt
Logically, the 1kg mass will take one half-life longer for all of the original atoms to decay. 

After one half-life, it will have the same number of atoms as the .5kg mass, and will clearly decay at the same rate as the other .5kg mass did originally. 

However, at the end of the decay period, we are talking about individual atoms and probabilities, so it really isn't possible to say exactly what will happen. 

A thought experiment - divide the first mass in two parts at the start of the experiment. Then you have three identical masses. They will all decay at the same rate, correct?

Where will the last original atom be? With two thirds probability, in one of the masses that was originally part of the larger mass. How long will that atom take to decay? Hard to say :)