Obviously, with almost 7 billion people on Earth, there is never a time when a person is not urinating. What I wanted to figure out is when it started.
My students and I tried to figure this one out, but none of us are great at history or math. Here’s the best we could do, using lots of conjecture and guessing.
Assuming that young adults and adults, on average, pee four times per day, with each experience lasting about 10 seconds. That means, it would take 2,160 people at a minimum for this to happen unintentionally.
There is a very slim chance of this happening. We estimated that it’s a .001% chance of it happening with that number of people.
We imagined a large stadium worth of people and assumed it’s safe to say that in an hour’s time, there will always be someone peeing. If you took a stadium of 60,000 and put one in each time zone, that may make it close to a reality. That means about 1.5 million people, about the population of Philadelphia, to make that happen. That puts the odds of 2,160 people doing it at .0015%.
The population of the Earth was about 1 million in 10,000 BC and 5 million in 5,000 BC, then humanity guaranteed a continuous stream of urine in around 8500 BC. So, we’d be well over 10,000 years in to our stream.
So, with our severely flawed math, history, science, and statistics, that’s our guess. We’re obviously wrong. What would be a better guess?
You aren’t taking into consideration sleep, or other diurnal patterns. Not until humans had populated the globe, and thus had populations spread out enough that you could guarantee that people were not all asleep can you start to do simple probabilities. This issue may dominate the question. Not until you got humans on each continent can you reasonably cover the day, and the total population in the already populated areas most likely already exceeded the number needed to cover all waking hours. The population needed to cover 24 hours just with occasional night-time micturation will be significantly bigger. So things are generally a bit more complicated.
According to historical sources we consulted, humans were already spread across the entire globe around 10,000 years ago and many cultures were beginning to use agriculture around this time. Both the Americas and all of Europe and Asia, in addition to Australia were populated then.
Granted, density was still probably pretty skewed.
If you assume the average person pees for 1 minute per day and it’s evenly distributed, probabilistically, with a population of 50,000 people, there’s 259 milliseconds in 10,000 years where someone is not peeing.
Given the probability rapidly diminishes as population size grows, I’m going to guess shortly after the Toba Catastrophe
Which would be when the first of the 24-hour news networks was established
More seriously, I’d use something like 20 seconds at least 6 times a day. That number is probably low for me, but maybe I’m just drinking a whole lot more water than everyone else out there.
I’m not sure the night-time is such a big problem. You could do the calculation so that we have a distribution like 5 average urination events over 16 waking hours, with just 1 event during the 8 sleeping hours. Kids and older people seem to pee more often at night - some of them probably more than once - so you’d be averaging them with the others who are more likely to make it all the way through the night.
That means you have a lot of little mini-problems to solve for various hours of the day, but you could focus on the hourly periods with the least people awake.
At the very least, I think you can conclude that your conditions are met once man reached North America. The consensus on that seems to be around 14,000 years ago, but there’s some dispute. (For example, this article says it could have been as long as 50,000 years ago.) By that point, not only did we have healthy population numbers, but good coverage of the time zones.
I like this. It’s probably good to consider the times when the least people are awake as part of the base number when thinking in terms of ancient history. 14,000 years ago seems pretty reasonable if that is when most people believe humanity was established in the Americas.
This is the point I was making. I suspect that the population in any given continent quickly rose to the point where for their daylight hours your continuum was satisfied. I suspect that that occurred pretty much as soon as our species evolved. So the question becomes - did the population reach a level high enough to reach the 24 hour continuum whilst only living in Africa, or did it need the spread of population to other continents. I suspect the former - that it occurred before the early diaspora - but if so, the calculation of needed populace needs inflating enough to cope with sleep times. Otherwise the point of continuum occurs coincident with a worldwide spread, or at a point somewhere in between.
If night-time peeing is 1/20 of daytime peeing (including just after sunset or just before sunrise) then 1 million sleeping humans should be able pee continuously without colonizing North America. Near-dusk will occur in West Africa and East Australia. I’m extrapolating from Shalmanese’s estimate.
Add 10,000 to the latest Australian estimate, and I’d say that if we don’t need North America, then 40,000BCE should work. Comments and further tweaking are welcome.
ETA: 7.
The patterns and frequency of diurnal urination were studied in 412 male and 244 female participants in the Baltimore Longitudinal Study of Aging. Men urinated less often (mean 4.8 voids per day) than women (mean 5.6 voids per day). There was no main effect for age decade. However, an age by sex interaction showed that voiding in men increased across the decades and in women it decreased. The highest frequency was found in women in their 30s (mean 6.2) and the lowest was in men in their 30s (mean 4.3). Men and women in the 70 and older age group had the same voiding frequency (mean 5.0). Normative patterns of diurnal urination across 6 age decades - PubMed
Older people and babies pee during the night, and sometimes several times per night. I don’t think it would take all that large a population in any one area to result in someone peeing somewhere at any given moment. I would think a population in the thousands if not tens of thousands would accomplish this fairly easily.