So long as we have imprecise measurements and only a finite amount of matter and energy then time is not an issue because we eventually run out of unique (or distinguishable by any of our measurements) humans that can exist and we can just ask the nearly equivalent question: "what is the probability that I am in the last 95% of unique humans to ever exist?" At some point every possible human has been created and out of that huge finite number each one of us is an index. The question is whether the heat death stops the creation of humans in this universe before 20 times the current number of humans has been created (for the 95% argument).
However, the statement of the problem completely ignores prior probabilities. For instance it's possible that we now have more than 5% of the total human population alive right now (that would be true if only 100 billion have ever lived). That would mean that the oldest people alive right now have a 0 probability of being in the last 95%. Additionally, we have a very good idea of whether we are in the last 99.9999999% of humans who have ever lived; the last 7 babies baby born on Earth have a slightly positive probability, everyone else has a 0 probability. What is the probability that 7 more babies will be born, replacing the current 7 who have the possibility of being the last? 1 for all practical purposes. 100 babies? Same. In fact it probably already happened while I was writing this comment. A billion? Probably still very, very close to 1. The calculation can be extended into the future based on our knowledge of our environment and society and the probability of existential risk. Unless we have a good reason for predicting the destruction of all humans we shouldn't predict that we are in the last 95% of unique humans. Humans are built to reproduce, they have an environment that will probably last thousands if not hundreds of thousands of years, and they are learning to build their own environments anywhere they want. These facts are far more important than which particular brain is randomly selected to think of the doomsday argument.
EDIT: The part about the last 95% of humans is wrong if there are more than 5% still alive; it would have to be some of the first 5% still alive for the argument to be wrong for them today.
only a finite amount of matter and energy
This is an assumption based on our current level of science and technology and failure of imagination. There are many possible ways around it: baby universes, false vacuum decay, even possibly conversion of dark energy (which appears to be inexhaustible) to normal matter, to name a few.
However, the statement of the problem completely ignores prior probabilities
I was unable to follow your logic there... Are you saying that the Doomsday argument is wrong or that it is irrelevant or what?
Edit: the argument is presented more clearly in a subsequent post.
Many thanks to Paul Almond for developing the initial form of this argument.
It is well known in these circles that the self-sampling assumption (SSA) leads to the doomsday argument. The self-indication assumption (SIA) was developed to counter the doomsday argument. This is a old debate; but what is interesting is that SIA has its own doomsday argument - of a rather interesting and different form.
To see this, let's model the population of a planet somewhat like Earth. From century to century, the planet's population can increase, decrease or stay the same with equal probability. If it increase, it will increase by one billion two thirds of the time, and by two billion one third of the time - and the same for decreases (if it overshoots zero, it stops at zero). Hence, each year, the probability of population change is:
During the century of the Three Lice, there were 3 billion people on the planet. Two centuries later, during the century of the Anchovy, there will still be 3 billion people on the planet. If you were alive on this planet during the intermediate century (the century of the Fruitbat), and knew those two facts, what would your estimate be for the current population?
From the outside, this is easy. The most likely answer if that there is still 3 billion in the intermediate century, which happens with probability 9/19 (= (3/9)*(3/9), renormalised). But there can also be 4 or 2 billion, with probabilities 4/19 each, or 5 or 1 billion, with probabilities 1/19 each. The expected population is 3 billion, as expected.
Now let's hit this with SIA. This weighs the populations by their sizes, changing the probabilities to 5/57, 16/57, 27/57, 8/57and 1/57, for populations of five, four, three, two and one billion respectively. Larger populations are hence more likely; the expected population is about 3.28 billion.
(For those of you curious about what SSA says, that depends on the reference class. For the reference class of people alive during the century of the Fruitbat, it gives the same answer as the outside answer. As the reference class increases, it moves closer to SIA.)
SIA doomsday
So SIA tells us that we should expect a spike during the current century - and hence a likely decline into the next century. The exact numbers are not important: if we know the population before our current time and the population after, then SIA implies that the current population should be above the trendline. Hence (it seems) that SIA predicts a decline from our current population (or a least a decline from the current trendline) - a doomsday argument.
Those who enjoy anthropic reasoning can take a moment to see what is misleading about that statement. Go on, do it.
Go on.
Still there? Then you've certainly already solved the problem, and are just reading to check that I got it too, and compare stylistic notes. Then for you (and for those lazy ones who've peaked ahead), here is the answer:
It's one of those strange things that happen when you combine probabilities and expectations, like the fact that E(X/Y)>1 does not imply that E(X)>E(Y). Here, the issue is that:
Confused? Let's go back to our previous problem, still fix the past population at 3 billion, and let the future population vary. As we've seen, if the future population was 3 billion, SIA would boost the probability of an above-trendline present population; so, for instance, 3-5-3 is more likely among the 3-?-3 than it would be without SIA.
But now consider what would happen if the future population was 7 billion - if we were in 3-?-7. In this case, there is only one possibility in our model, namely 3-5-7. So large present populations are relatively more likely if the future population is large - and SIA makes large present populations more likely. And this removes the effects of the SIA doomsday.
To sumarise: