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No it doesn't depend on a multiverse at all. (Does all statistical reasoning require multiverses to exist?) It only requires a belief that the universe we see is one example of the set of all possible (imaginable) universes.
You don't seem to be understanding the implications of the anthropic principle. For every human that is conceived, there are billions of sperm which are thrown out, and there are even more (billions upon billions upon billions(*A)) of potential genetic combinations which could happen but don't. Had the sperm which became you not fertilized the egg which became you, you would just not exist to have this conversation. Full stop. The probability that you in particular would exist is just too small. The number of possible humans is of similar magnitude to the number of atoms in the universe.
(A) The human genome has 3,054,815,472 base pairs, and two random humans might differ by up to 0.6% of their genome, so we can say that two random humans are separated by about 18M base pairs, whereas the rate of mutation in human DNA is ~2.5×10^(−8) per base per generation, so each human will have about 76 new mutations. To simplify a bit (assuming mutations are evenly distributed, which they are not, but you will see that it doesn't matter), this means the possible range of single mutations that are still considered human is bounded between 4^76 = 5.7 x 10^45 (all mutations between humans occur on the same 76 nucleobases) and 18M choose 764 (any of 76 mutations could occur anywhere in the genome). Either way, the number of possible humans is really big, so if you were not born when you were born, you would never have been, at least not in our light cone.
This was addressed by ResoluteRaven below. The answer appears to be "not very hard", and that was also the gist of the paper I linked above about combining HCN and H2O to make amino acids.
This is poor logic, because it can be continued ad infinitem to explain anything inconvenient for your position, and makes your position unfalsifiable. (You might say it proves too much. If your position is unfalsifiable, then it is not testable. To put it another way, suppose that 50 years from now scientists were to demonstrate abiogenesis in the lab. You could still argue that they merely discovered the method by which God created life. How convenient, considering it would also the method by which life could have arisen without a God at all.
In the modern day, I've heard creationists arguing on behalf of the position that the earth is only 6,000 years old. When confronted with the fact that fossils can be dated to millions of years ago, they fall back to the argument that if fossils appear to have been buried for millions of years, they must have been placed in the rock formation by an intelligent designer to appear that way, so as to trick modern-day humans. This argument can of course be extended to argue that everything before any arbitrary moment in the past has been retconned, and God just created a world to look convincingly old. If your Designer is all-powerful, I guess that might make sense to you, but it is equally valid to suppose that the Designer didn't do much more than set some parameters on the Big Bang and press a button to see what would happen.
I hate to be trite, but another commenter below has explained this already. I may suggest that if this topic matters to you (or is truly critical to maintaining your faith), then you try reading the first few chapters of a textbook on molecular biology for the relevant background. You don't have to read very far. I got to chapter 3.
I'm really sorry, but my friend has spent an inordinate amount of their career arguing against intelligent design. Probably about as much time as they have spent doing biological research. Given that the return to humanity is much higher if they spend their time doing research, I really don't want to provide them access to more ideas from intelligent design.
It does require it, I think.
One draw from a haystack vs 100000000000000 draws will have different chances of hitting the needle.
If only one universe exists, and most possible universes are very non-conducive to life, it should be surprising to us that we exist, since that seems so unlikely. At that point, we should be looking for explanations that might make it more likely, like multiverses or theism, or it being necessary that the universe be that way, or actually, most universes are conducive to life after all. But we can't just say that in worlds where we woke up it would look like worlds where we might be able to wake up, because the really surprising thing here isn't that but why the hell did we wake up at all, if we are indeed in the only universe, which should by every expectation be very hostile to life. (note, I'm assuming those two things, not asserting them here)
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