You may have seen an argument online known as the "doomsday argument." It uses mathematics (specifically probability and statistics) to argue that humanity will likely come to an end within the next several thousand years. This page first presents the argument without comment, and then analyzes the argument and discuss its shortcomings.

Say that you have a box in front of you that contains a number of paper slips, numbered consecutively from 1. You are told that there could be any number of slips inside; however, you are not able to look inside to count or estimate how many there are. You can, however, draw one slip of paper out of the box and read the number on it. If the number on the slip of paper is 5, the box could still have any number of slips of paper in it (at least any number ≥ 5). However, you would probably think it to be more likely that that the box contains 10, or 50, or some relatively small number of slips of paper than, say, 1,000,000 slips of paper.

We could look at the human race in a similar manner. We don't know exactly how many humans there have ever been (we're missing a lot of information about birth and death rates in history, and the answer also depends on when we start counting people as "human") but we could estimate that there have been somewhere between 70 billion and 120 billion humans throughout all time.

Now, suppose that we want to know the total number of humans that there will ever be. Considering the example above, say that we reached into our box above and drew out the number 120 billion. Now, the figure of 120 billion could represent 0% of the total slips in the box (in which case the number of slips is infinite), or it could represent 100% of the total slips in the box (in which case the number of slips is 120 billion), or any percentage in between. Since, for all we know, any percentage is equally likely, we could model this with a uniform distribution. Now, the probability that 120 billion represents between 5% to 100% of the slips in the box is 100% − 5% = 95%. So, there is a 95% probability that the total number of humans ever will be between (120 billion ÷ 100%) and (120 billion ÷ 5%), or between 120 billion and 2.4 trillion.

Now, if we assume that the world population stabilizes within a few decades to 10 billion and that the average human lifetime becomes 80 years, then (2.4 trillion − 120 billion) humans will be born within the next 18,240 years. Therefore, there is a 95% chance that humanity will last no longer than 18,240 years.

The first argument presented above, regarding the piece of paper, is a valid one. It could be justified using Bayesian probability. The analogy is also reasonable; if we know that there have been 120 billion humans, it's probably reasonable, basing our argument solely on that number, that the number won't be significantly larger than that.

The argument then uses a uniform probability distribution to model the probability to how far along, percentage-wise, the human race is. This might seem like using statistics to turn vague ideas into impressive-sounding ideas, but using a uniform distribution is a valid way to model our ignorance of the phenomenon. The only problem is that, with only a single observation, and without using any information that we might know about populations, any estimate that we make is going to be rather rough.

For example, consider that we were estimating the lifetime of a person. If we knew nothing about human lifetimes, then we have no idea whether that person's life is probably 1% over or 99% over or somewhere in between. We could model this lack of knowledge using a uniform distribution. According to this distribution, there is a 95% chance that the person has lived between 97.5% and 2.5% of their life. If our person were 40 years of age, then 95% chance of living somewhere between approximately 41 and 1,600 years of age. In reality probably something like 99% of people will live to be between 41 and 110 years old, so it's not bad for a guess, but far from perfect due to our lack of information. If our person were 1 year of age, our model would give a 95% chance of that person living to somewhere between approximately 1.025 and 40 years of age. If our 1-year-old is in a developed country, this doesn't seem right; we would expect a life expectancy of 70 years of age. For a 100-year-old, our model would give a 95% chance of living to between 102.5 and 4,000 years of age, which doesn't seem right either; we would expect much more than 5% of 100-year olds to die within the next 2.5 years.

Looking at the above examples, we see that the uniform probability distribution is an okay, if rather rough, model for a middle-aged person, but not for a very young person. Looking at the human race, is the human race at the start of its lifecycle, in the middle, or near the end? We don't know. However, without knowing that information it's difficult to say whether the estimates that this model gives are reasonable at all.

Let's assume that the model is reasonable, and that there is a 95% chance that there will be less than 2.4 trillion humans. Now let's look at the final paragraph of the argument:

Now, if we assume that the world population stabilizes within a few decades to 10 billion and that the average human lifetime becomes 80 years, then (2.4 trillion − 120 billion) humans will be born within the next 18,240 years. Therefore, there is a 95% chance that humanity will last no longer than 18,240 years.

This portion of the argument is incorrect. The rest of the argument assumes that we have no idea about how the human population is distributed, but this paragraph assumes that we do. Should we not take into consideration everything we know about the human race in the rest of the argument? Should we not take into consideration the fact that humanity has been around from 50,000 to 200,000 years (depending on when you start counting people as "human")? Would it not make sense that, if the human race took 50,000 years to grow to its current size, that it would take 50,000 years to dwindle to zero? Should we take into consideration the fact that the human race is growing exponentially and so is probably still "young" in terms of the lifetime of a species? There are many factors that affect the lifetime of the human race, and without taking them into account, any estimate of its lifetime is probably going to be very inaccurate.