I read a paper - two papers actually first by diaconnis et Al and followed up with J. B. Keller. These papers debated on the probabilistic nature of the coin toss of which both argued against it.
Diaconnis group found out that, the toss of a coin isn't really 50/50 as we've been taught in school. They proved that, the probability is rather 49/51 and it's biased towards the side facing up. This was achieved with experiment.
In Keller's approach , which was a theoretical one and preceeded Diaconnis, he proved that, if you know the initial conditions of a coin toss, then, you can predict the out come of a coin toss.
I questioned how can you find this outcome of a toss? I had an answer which I'll share in no time.
We propose a kind of toss to use in this theoretical exploration. In our coin toss, the force applied to the coin is perpendicular to the coin and the coin takes off horizontally straight without deviation and follows the same path down at the same time, the same force causes it to rotate about the y given the orientation in figure 1.
We know already that, the coin has 3 areas. Two faces and one side. The heads and tail and the side of the coin. If the coin is tossed and we view from birds eye view, we see that, the coin's parts transitions through heads, tail and side.
Also, before the toss, the coin was having one side facing up and after the toss, will have a side facing up.
This is akin to a cosine graph where we have 1, 0 and -1. And the curve transitions through these numbers till it stops and settles at any of the values depending on the x value.
In the case of the coin, we can let the sides be the values 1 or -1 and the side be zero. The reason being that, the sides are the outcome and zero has no outcome which corresponds to the side.
For the rest of the values, cosine begins with 1 at zero - no activity this corresponds to the side that faces upwards if the coin is about to be tossed. And the other side, the -1. This means 1 can take any face but once a face has being assigned, the -1 takes the other face.
Now we shift to using 1, 0 and -1 in our analysis with the assumption 1 is heads and -1 is tail.
If the coin is tossed, it transitions between these values till stopped and a final value is obtained. If it's 1, it's heads if 0 it landed on its side and if -1, it's tails.
Now, we have for find a way to track the values. We have two ways.
The best is to track the angular displacement of the coin because cosine takes angular values so we find the angular displacement of the coin.
We assumed that, the force we used to move the coin up is equal to the force we used to get the coin rotating.
We designate the earlier \[F_T\]
And the latter \[F_R\]
Further analysis, \(F_T\) has to be \(2mgR\) which arises from the integral of \(mg\) as the coin goes up and dowm so we integrate over \(-g\) and \(g\)
We also have \(F_R\) to be \[\frac{4I\theta}{R^2t^2}\]
Combining both with the moment of inertia of the coin, which is \[I = \frac{mR^2}{4}\]
We obtain \[\theta = \frac{2gt^2}{R}\]
Anyways He exist in a space. Whether this space can be created is a problem but i have an interesting answer.
Which makes time the only controllable thing here.
We can plug this in our function to get the equation \[\Gamma = cos \bigg(\frac{2gt^2}{R}\bigg)\]
Which is for the outcome of the coin.
So if we obtain \[ \Gamma = \begin{cases} 1, heads \\ 0, sides \\ -1, tails \end{cases} \]
But the question is what if it doesn't land exactly on heads but tilts a bit. Here we assume no bounces. If we assume no bounces, if the coin lands, it will topple over and land on the other side outside the point of contact to the side of the tilt never away from there because that's where the center of mass of the coin lies.
So still the same. The coin will land on the side which is facing up at the point of contact so we can generalize out conditions that, if we obtain an even number, heads and odd number tails and a number 0 or very close, sides.
The outcome of a coin toss can indeed be predicted with certainty if the initial conditions are known with certainty. Given this equation for the outcome of a coin toss, the coin is destined to land the same way as it begun which makes Diaconnis investigation partially true maybe the other approximately half was due to human error or errors in the measuring apparatus.