The fact that you've gone through the trouble to do the math on this is both mind boggling and fantastic!Great question! Here are some ponderings.
Let's assume a knotting takes 15 minutes.
With 24 hours a day, it would only need 96 knottings that are perfectly coordinated one after the other in order to have constant knottings around the clock.
Since people and dogs don't coordinate sex like that, let's assume knottings happen at random over the day. Of course that's a big assumption due to how the population of the world is unequally distributed and there are times when a bigger portion of the world population sleeps or works than others, but anyway ... How many random knottings do we expect to need so that the whole day is covered?
I'm not sure how to calculate this, but there's a similar problem called the coupon collector's problem where we deal with a set of coupons that we want to complete instead of a continuous span of time which we want to cover. To get a rough idea of what we are dealing with, we can turn our continuous problem into a discrete one by splitting the day into slots – each slot would correspond to a coupon we want to collect then.
If a knotting takes 15 minutes, then how long would each slot have to be to make sure that knottings in subsequent slots overlap? Well, if one knotting starts at the very beginning of a slot and the next knotting at the very end of the next slot, then the time between those knottings is basically two slots long. In order for there being no pause between knottings, the first knotting must last two slots. So we make a slot last half a knotting, 7.5 minutes.
This means we need 192 slots which correspond to 192 coupons a day. Then we can use the formula for the coupon collector's problem to calculate the expected number of knottings needed so that each slot gets used, which in turn means that there's surely no second of the day without knotting. The Wikipedia article gives the formula as approximately ...
E(knottings) ≈ 192 × log (192) + γ × 192 + ½
... where γ is the Euler–Mascheroni constant at approximately 0.5772156649 and "log" is the natural logarithm, i.e. the logarithm to base Euler's number e. Two constants named after Euler in one equation ... we need to be a little careful there. Anyway, the result of the computation is E(knottings) ≈ 1121. So we expect that it needs on average 1121 randomly happening knottings a day to have the whole day completely covered with knottings.
We could calculate more, e.g. how many knottings it needs on a particular day to be 99% certain that the whole day is covered, not just what it needs on average. But let's stop there and ask ourselves whether 1121 knottings a day is realistic to happen. The human world population is over 7,800,000,000 right now. If we divide that number by the required knottings, we get that we would only need one in about seven million people to have a knotting on a particular day. That's about the rate of a single knotting in Tennessee or a single knotting in Bulgaria in a day.
Sounds realistic!
Don't hit me, if I've made a mistake. This is not a scientific paper, but a quick educated estimate.
Corrections are welcome.
Paul Erdös doesn't happen to be in your family tree I wonder.? Well done.Great question! Here are some ponderings.
Let's assume a knotting takes 15 minutes.
With 24 hours a day, it would only need 96 knottings that are perfectly coordinated one after the other in order to have constant knottings around the clock.
Since people and dogs don't coordinate sex like that, let's assume knottings happen at random over the day. Of course that's a big assumption due to how the population of the world is unequally distributed and there are times when a bigger portion of the world population sleeps or works than others, but anyway ... How many random knottings do we expect to need so that the whole day is covered?
I'm not sure how to calculate this, but there's a similar problem called the coupon collector's problem where we deal with a set of coupons that we want to complete instead of a continuous span of time which we want to cover. To get a rough idea of what we are dealing with, we can turn our continuous problem into a discrete one by splitting the day into slots – each slot would correspond to a coupon we want to collect then.
If a knotting takes 15 minutes, then how long would each slot have to be to make sure that knottings in subsequent slots overlap? Well, if one knotting starts at the very beginning of a slot and the next knotting at the very end of the next slot, then the time between those knottings is basically two slots long. In order for there being no pause between knottings, the first knotting must last two slots. So we make a slot last half a knotting, 7.5 minutes.
This means we need 192 slots which correspond to 192 coupons a day. Then we can use the formula for the coupon collector's problem to calculate the expected number of knottings needed so that each slot gets used, which in turn means that there's surely no second of the day without knotting. The Wikipedia article gives the formula as approximately ...
E(knottings) ≈ 192 × log (192) + γ × 192 + ½
... where γ is the Euler–Mascheroni constant at approximately 0.5772156649 and "log" is the natural logarithm, i.e. the logarithm to base Euler's number e. Two constants named after Euler in one equation ... we need to be a little careful there. Anyway, the result of the computation is E(knottings) ≈ 1121. So we expect that it needs on average 1121 randomly happening knottings a day to have the whole day completely covered with knottings.
We could calculate more, e.g. how many knottings it needs on a particular day to be 99% certain that the whole day is covered, not just what it needs on average. But let's stop there and ask ourselves whether 1121 knottings a day is realistic to happen. The human world population is over 7,800,000,000 right now. If we divide that number by the required knottings, we get that we would only need one in about seven million people to have a knotting on a particular day. That's about the rate of a single knotting in Tennessee or a single knotting in Bulgaria in a day.
Sounds realistic!
Don't hit me, if I've made a mistake. This is not a scientific paper, but a quick educated estimate.
Corrections are welcome.
and guess at around what time was this session recorded.
Love itView attachment 259253
Lets see how many knots you can count
