Today's actual, proper video is here:
The Doomsday Argument is a mathematical, or rather statistical, argument that we are likely to be in about the middle of the number of human beings who have ever or will ever exist. Due to exponential increase in population, however, that seems to imply that we are near the end of human history.
The argument is roughly that one's life can be considered to be at a random point in human history. Since for most of history, the world's population has been very small and only recently has our population started to grow, exponentially in fact, the probability that we are towards the end of history is much higher than if we are at the beginning. Even if we assume that we are only half way through the sequence of human births in history when we were born, considering that population has been doubling every thirty years, that gives us less than two centuries to go. If the population reaches a plateau of about 10 billion, that apparently gives us about nine millenia, though i can't see the maths there. That's not very long compared to the history of the species.
However, i think this argument is wrong. Firstly, it is really an argument about the probability that that thought will occur at a particular time, not that your birth will. Since most human beings die in infancy or childhood, most of them probably don't have that thought. Subtract those numbers from the population and you are left with a smaller number. It also seems to assume we will stay the same throughout history, i.e. that we will tend to have thoughts like that. That's not the case if the argument is refuted, we adopt a mindset which is incompatible with the argument or we become too stupid to understand it. It also won't work if we become immortal, or even if the future consists of a very small population persisting for many millenia.
Therefore, so far as i can see the argument doesn't work. Whereas it's possible that the human race is about to become extinct for a whole load of other reasons, this isn't one of them. However, note that several of my counter-arguments mean the future is pretty bleak even though we survive.
I think there's a place for the Doomsday Argument, but it isn't how it seems. It's more an argument that thoughts that the world or species might end soon will become rarer per head of the population. This could be because some kind of process will take place which reduces the number of human heads, or it might just be that we're going to cheer up a lot in the future. I don't know, we'll have to wait and see. I suspect, in fact, that it's unprovable because if it turns out to be true, nobody will be around to know about it.