The original question was: From biology and genetics we know that any group of living organisms had a mitochondrial most recent common ancestor (mitochondrial Eve): a female organism who lived in the past such that all organisms in this group are her descendants.

The original question was: From biology and genetics we know that any group of living organisms had a mitochondrial most recent common ancestor (mitochondrial Eve): a female organism who lived in the past such that all organisms in this group are her descendants.
How can we [theoretically] prove this? (I.e. without assumption that there’s equal probability to mate for any two individuals).
Also: can we prove that one of 3 billion women currently living on Earth will be the only ancestor of all human population some day in the future, and all other currently living women (except her mother and daughters) will have no descendants at that day?
Physicist: The fact that everyone on Earth has a common female ancestor if you go back far enough is a direct consequence of the theory of common descent. It looks like everything that lives is part of the same very extended family tree with a last universal common ancestor at its base. In order to have two familial lines that never combine in the past you’d need to have more than one starting point for life, and all the evidence to date implies that there’s just the one. Luckily, you don’t have to go all the way back to slime molds to find common ancestors for all humans; the most recent were standard, off-the-shelf people.
It turns out that animal and plant cells aren’t particularly good at producing usable energy, so before we could get around to the business of existing we needed to get past that problem. The solution: fill our cells with a couple thousand symbiotic bacteria. Literally, they’re not human; mitochondria reproduce on their own and have their own genetic code. There’s a hell of a lot of communication and exchange of material between them and our cells, and without them there wouldn’t be an us, but they are (arguably) separate organisms that we are absolutely dependent on and which are completely dependent on us.
Here comes the important bit: eggs cells have mitochondria but sperm cells don’t, so mitochondria are passed strictly from mother to child. There’s no implicit reason for your mitochondria and your father’s to be related at all. The nice thing about that is that it keeps the genetic lineage very simple: all of your mitochondria are essentially clones of those in the egg cell you started as and (for our female readers) any of your children’s mitochondria will essentially be clones of yours. The genes of sexually reproducing beings are a lot trickier to keep track of over time; every generation half of our genes are dumped and the other half are shuffled with someone else’s (which makes your DNA is unique). The one real advantage to talking about mtDNA (mitochondrial DNA) is that you only have a single chain of ancestors to worry about. By the way: you can do exactly the same thing with the Y-chromosome and direct male lines.
If you have a group of creatures with two types of mitochondria, two “haplogroups“, living under a population ceiling, then eventually one or the other will be bred out. The math behind this is essentially the Drunkard’s Walk. The number of folk in a haplogroup can increase or decrease forever, unless it gets to zero; given nough time and no where else to go (a population limit), eventually the drunkard’s walk will take him off a cliff (zero population). So, if you start with a small village and several haplogroups, then after a few generations you’ll probably have fewer.
That isn’t saying much. It boils down to the rather fatalistic statement that “in order to be the last thing standing, you just have to wait for everything else to die off”. We can’t prove that a woman today will eventually be declared, very post-mortem, the Mitochondrial Eve to everyone (that is; all but one haplogroup will die off). But statistically: that’ll definitely happen. To within less than a 1% error, every inherited line of every kind has died off; practically every species, sub-species, gene, haplogroup, whatever, has gone extinct leaving only the amazing scraps that remain. That’s evolution in a nutshell: you chip away all the life that isn’t an efficient, functioning organism (and then a hell of a lot more besides) and the inconceivably tiny fraction that remains is (some of the) efficient, functioning organisms. So, chances are that every living haplogroup presently around will go extinct eventually. When there’s one haplogroup left, then you can say that they all have a common Mitochondrial Eve and when there are zero haplogroups left, then all human issues become moot.
In order to definitely not get a modern Mitochondrial Eve, you’d need human populations that are absolutely independent (and viable) forever. Maybe if we colonized Mars and then completely forgot 