Watch this video from numberphile.
This video explains why almost all integers have the digit "3" in them.
http://www.numberphile.com/videos/three.html
Really, watch the video. I'm going to explain it below, but the video will probably make it more clear, and you need to understand that video, i think, in order to understand what i say next.
so the equation is this:
T=1-(9/10)nth
(sorry, hard to write equations in the forum editor)
T is the number of integers which contain "3"
1= 100 percent of the numbers you are condisering
N = the number of digits in the total number, so for 1000 N=4
and there's a 9/10 chance of NOT getting a 3 in any digit place.
What this equation says is that there's a 90% chance that each digit won't be a three, but a 10% chance that it will. For each digit you add the chance there will be NO 3's has to be multiplied by 90%. so for two digits, it's 90% of 90%, or a slightly greater chance of there being a 3.
The more digits you add to the integer the closer T gets to 100% until almost every number there is has a 3 in it.
This same structure works for why time doesn't go backwards. For time to go backwards a particle has to go from A to B, and then retrace itself back to A.
For any particular particle the same rules apply. For simplicity, lets say there are ten directions the particle could go in the next instant (in reality there are far more which would make the ultimate numbers i come up with here a very conservatively small number in comparison), and one of those options is back to A, where it came from. Going back to where it was is just as likely in this scenario as any other direction with a 10% chance of happening.
Just like when adding more digits to that integer adding another particle means you have to multiply the odds together. The difference between the two outcomes is that in the first instance we were looking for a number where ANY digit was 3. In the case of time going backwards, we are in essence looking for the number where ALL digits are 3. They all go back to where they started, in other words.
So if each particle had ten directions it could go, with one direction being back where it started, with two particles, the chance of going back in time is 1/100 from moment to moment. But for time to truly go backward, not just 2 particles have to go back where they were.
Not just a million particles(1/10to the -6 chance of going back in time), not just a one in a trillion particles (1/10 to the -12 chance) but all particles in the universe (estimated to be about 1/10 to the -82 atoms), not even counting all the bosons and virtual particles that would ALSO have to go backward which means that 10 to the 82 is a woefully small number compared to what would actually be involved, and that's to go backward in time to the immediately proceeding instant.
To go back to the instant immediately proceeding probably means the smallest time length i know of which is the plank time. That's the time taken for a massless particle to travel the plank length. So to my estimate that means to go back in time one 10 to the -44th of a second there is (extremely generously, considering i started with only ten directions a particle could go, then called the whole nucleus and electron arrangement one particles and left out bosons, virtual particles and whatever the hell dark matter and dark energy and space are...) a 1/10 to the -82 chance of that happening randomly. And that's assuming all paths are as likely.
Want to see what that looks like?
0.0000000000000000000000000000000000000000000000000000000000000000000000000000000001%
That's the chance time will go backwards one plank time from these numbers. As i said, there are probably a lot more zeros involved when you take into account all the things i was generous about.
(What about the odds of winning the michigan lottery?
0.000011%)
And to go back two plank lengths, you have to square that probability... you see where this is going, right?
And that is why something ELSE happens... time goes forward.
What's interesting is that by these statistics any SINGLE future is just as unlikly as going backward in time. If you picked one randomly, you would almost certainly be wrong to the same tune that almost every number has a 3 in it. But in essence almost 100% of all options is not the past, so no matter which of the futures does occur it will be from amongst this vast range of improbbable futures that it will be selected.
That's just using the numbers, but in real life, there's a pretty good reason that balls roll down hill, and not up, and that is the forces acting upon it, and inertia, which are not accounted for in my example, and those are additional compulsions toward not repeating the past.
The point of this is to show how statistics helps grasp entropy, and i think is a nice qualitative look at how time flows in one direction.