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The typical argument utilizes Drake's equation. The problem though is that the size of the Universe (vast) is only part of the equation. The other part is the odds per planet of life developing and no one knows anything about that at all. Is it one in ten. Is it one in one billion? Is it one divided by the number of planets in the Universe?
One might ask where would life come from. You might then speculate that abiogenesis is the answer. Molecules combine in such a way that a self-replicating molecule is created. The odds of this, though, are apparently extremely low. Some have calculated it as to be so low that it is not even possible. This is a typical argument used by creationists. I don't agree with the creationists' "not possible" but they might be onto something. It might be so low as to only have happened once. Or then again, maybe there is some as yet undiscovered self-organization which makes it quite likely.
One should also be open to the idea that there is no other life in the Universe.
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