Mmm. It seems to me the problem with this argument, like most, lies in its assumption
s, and not merely the ones about probability mechanics
One: "human growth is geometric"1. There's no reason why population must grow, only the fact that it has for most of human history. And there's certainly been periods where the population of a country has shrunk due to high death rate and low birth rate caused by human and natural crisis - the Greek Dark Ages as well as the fallout of the end of the Western Roman Empire are two examples. Expand this to wide enough an area and the lowering of human population is certainly possible. Let's call on Malthus. He realized disease and war were perfect population controls, but that doesn't require them to be extinction level events. Epidemics and limited conflict wars are more likely than pandemics and nuclear war, and even serious nuclear war does not rule out the existence of survivors. Just because something hasn't happened before is not proof that it won't happen in the future. For a more futuristic way, imagine a world government that enforced strict reproductive restraints ala the UN and ARM in Larry Niven's Known Space universe. Alternatively, imagine a world where people increasingly chose to upload themselves in a destructive way - much of the population would disappear into machine consciousness. Continual population growth is not guaranteed, so it's possible a very large amount of people could be born and die between now and doomsday.
Two: "the amount of people increases towards the end of the
graph". He seems to presuppose a swift and utter end to humanity by this statement. If humanity becomes extinct, there's no reason it couldn't occur in a gradual decline. If we have really poisoned the planet to the point of no return as some fanatic environmentalists have claimed, our decline could take hundreds of years as the human race slowly became sicker and unable to survive on the increasingly environmentally damaged planet. 2 Returning to the second scenario of future population control, the entire human race could slowly become jupiter brains and the like over millenia as the cautious decide to finally take the leap and the technophobic finally die out. And once humanity expands beyond our solar system, anything goes. Even if intrinsic human frailities must lead to the deaths of civilizations in the long term, other colonies would survive in a frontier state, staving off extinction. 3 How long could the human race move from planet to planet, losing a culture every millenia or so, until it finally collapsed? Expand out any of these possibilities and no doubt many more I've failed to think of, and the end of the human race truly could come with a whimper, not a bang.
For these two reasons, I find it hard to take this argument serious. It's a possibility, but so is the possibility of continued human existence until the end of the universe by heat death or gravitational collapse. And given enough time and technological improvement, I really don't see why it all has to end there either.
1: I'm tempted to point out that if he represents human population as always growing, then that means as a mathematical function it can't go to zero (become extinct) since it only goes up, but enough deliberate misunderstanding.
2: For another Earth First-ish scenario, the voluntary human extinct movement could succeed in slowly convincing everyone to stop breeding, but I doubt it.
3 A Deepness in the Sky by Vernor Vinge has an excellent representation of this idea in the debate between Pham Nuwen and his wife. I side with Pham Nuwen - I refuse to believe that progress beyond anything is impossible, except when you think it is and refuse to try.