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Googling: the Doomsday argument.

In short, the essence is as follows. Let N be the number of all people born in the past since the appearance of civilization (t) and up to some point in the future (T). We don't know exactly where the current moment is on this t-T axis, but the probability that we are somewhere at the very beginning of this segment is not too high. To understand why this is so, consider a real historical example.

During the Second World War, the British estimated the number of tanks the Germans had by the serial numbers of damaged vehicles. For example, if the serial number of a tank is 50, then the probability that the enemy has more than a thousand tanks is less than 5%. Because the probability of getting into the first 50 tanks built out of 1000 is only 5%.

By reasoning in a similar way, we can estimate the probability of where we are on the t-T axis. For example, if my sequence number in the list of all people ever born is 50 billion, then there is a 95% probability that the number N – i.e., the total number of all people ever born-will not exceed one trillion (50 billion * 20). Well, it's just extremely unlikely that I will be in the top 5% of all people ever born (see the example with tanks above).

From here, you can estimate the value of T – the very Day of Judgment. At the moment, if we substitute the current level of population and growth in these calculations, the time T will not exceed 10 thousand years. I.e., with a 95% probability, humanity will completely die out within the next 10 thousand years.

However, the good thing about probabilities is that they are only probabilities. Of course, there are no guarantees that we will not become extinct in 15 years or not at all.