After completing an essay for an elective in University, I checked my word count to find it was an exact multiple of 100. For a moment I thought to myself, what a remarkable coincidence, what are the odds of that! hmm... what are the odds of that. I've written many papers, is it still unlikely that at least one of them would have a word count that is a multiple of 100? Or is it in fact likely?
You are a magician.
A bad one. In an attempt to impress people at parties you attempt to guess cards they pick at random from a deck. Each time you try it's unlikely you will guess correctly: 1 in 52.
So most people will identify you are a bad magician and probably avoid you. Your strategy is that eventually after enough attempts you will get it right and impress someone, they will become your friend, maybe you can quit magic etc.
If you try it 52 times a night every night, on average you will get one correct guess a night. But you do not have a 100% chance of getting one correct guess every night, there is always a chance you will go home a failure, and there is always a chance you will guess correctly every time. This difference is important!
Things are amazing or horrible. Impressing no one and going home alone - horrible. Impressing any number of people - amazing. So you might want to know how many times you need to try the trick to get some percentage chance of going home happy - ie. getting one or more guesses right. You know that you will never get to 100% confidence of going home happy -
due to this being a random event, with a chance of total failure every night.
Maybe you would like to have a 50% of going home happy every night. The plot below shows experimental data about the odds of getting more than 0 guesses right after a given number of attempts.
CRITICAL: all we care about is that we impressed at least one person. Impressing 1,2,3 or 50 people is all treated the same: not crying yourself to sleep.
After 36 attempts, the probability is 0.5 you will go home happy. I used the average of 3000 trials to generate each point, but there is still reasonable noise from the random data.
We have shown that once you have written 69 or more essays, (papers etc) the odds are over 50% that one of them will have an even multiple of 100 words. Adding to this that the number of words in a document are likely not random if there is a minimum word count, it is probably clustered around multiples of 100 (depending on how lazy you are). So the paper I wrote was likely not special or magical, and if I had spent less time determining this I would likely have proof read it more thoroughly.
We have seen that if you have written 69 or more essays, papers, etc. there is a greater than 50% chance that you have written one or more with an even multiple of 100 words. Likely, the distribution of essay lengths is not random, but rather clusters around multiples of 100 (assuming your are busy or... lazy). So this is to say, that my essay was not special, or magical - and had I decided this after writing it, I would have saved much math analysis, and likely had time to proofread the paper before submitting it for a mediocre grade.
This turns out to be a scaled geometric distribution. If we said that we cared about the odds of x failures followed by one success it would be exactly geometric - and would simply be multiplied by 1/x. - For our purposes this is mostly irrelevant. Do not remember this.
With this groundwork laid, lets take a look again at the essay example with a 1 in 100 chance of "success" because the numbers are nicer to work with. How many essays would I have to write before there is a 50% chance one of them has a multiple of 100 words. Our intuition from earlier - knowing that 100 attempts does not give 100% chance of success - suggests that we will have to try more than 50 times.
I approached this by checking what the odds were that that I did not succeed. This is easier as there is only one route to failure to check. The odds of failure are 0.99 each attempt. The probability of an event occurring n times consecutively can be given by P^n, in this example 0.99^n.
This leaves us with the solution to this case 68.968 attempts. Interesting. Can we generalize this for other cases? It would be nice to know some factor I could multiply by the odds as an offhand calculation to find out when something would be likely to occur. For this example, if the odds are 1 in M (here M=100), we must attempt the task 0.6897*M times.
What if the odds were different, for the card example we saw before it was 35.7 attempts out of 52, giving 0.686*M. What if the odds were say 1 in 1000, would we need to attempt 0.686*1000 times? For this we will need to generalize the formula as the number of attempts goes to infinity.
The proof for this I will try to present legibly in another post. While not onerous, it is slightly tedious in keeping track of notation, and deserves it's own article.
Considering the feasibility of a time accelerator
I've often considered the possibility of accelerating someone to relativistic speeds in order to slow their experience of time relative to the rest of earth. The serious issue would seem to be that it would require huge amounts of energy to accelerate to those speeds, and even worse, to slow down and come back to earth.
A simplified solution might be to revolve someone, ideally a pair of someones (this is the kind of trip best taken with a friend) up to relativist speeds. This gives the advantage that you do not have to travel as far from earth, and supplies could be delivered to the center of revolution to some kind of tether keeping the two objects together.
So lets check the math.
In order for this trip to be a decent use of time, you need a good return on your invested time. I would want at least a ten times speed acceleration. Trade 3 years to skip 30 earth years. So this sets our desired speed.
Our other constraint on this journey is the maximum force exerted on a travelers body. Since this will be a long trip, I will set this at 3G - 3x earth's gravity, which will probably still suck for your internal organs.
So how fast do we need to be moving to get this back to the future-esque effect?
Damn. 0.995c is very close to the speed of light. For comparison, the crew of Apollo 10 reached almost 40000kph on their return voyage from the moon, the fastest any human has ever traveled. We want to travel about 27000 times that fast!
Lets get up to speed. Again, we will limit the acceleration to 3G to make some attempt at keeping this passenger alive.
Total time to get up to speed is 117.5 days. Not bad! Kind of like a really long roller coaster ride.
Okay, we know what we have to do; accelerate someone up to .995 times the speed of light. This will require energy. Significant amounts of energy. Let's use the current state of the art, a high power ion engine. Their efficiency is around 60-80%, but lets call it a nice round 100% for now.
We can see that just the kinetic energy involved would be 4x10^20 J. That would require an average power input of 40 TeraWatts. Hmmmm, that's really getting up there. The average global power production is only 2.4 TW. Maybe you have some friends in high places that are willing to amp up global power 17 times and give you ALL OF IT. You only need it for 117 days right?
You borrowed the power, you know what to do with it and how long it will take. So how long does this tether between you and your friend need to be? It would be 6x10^15m or 0.6 light years across.
Shit, that's about 40 000 000 Astronomical Units (AU), which is 40 million times the distance from the earth to the sun. A 20cm cable would be about 5x10^13 cubic meters. But good news! there is in fact enough aluminum in the earth to make a cable that size!
Unfortunately, it looks like no one will be taking any trips to the future for at least a while. The size and power requirements of the device are immense, and it would take technology well beyond ours just to get the raw materials off of earth.
However it is something fun to think about. Perhaps some day we can overcome the maximum force issue that causes the device to be so large, and we will have new sources of power that will make these numbers seem smaller. Until then, this will remain a thought experiment.