Distribution result of flipping coin with same initial conditions repeteadly

Question

Still related with that question Flipping a coin with same initial conditions.

While it was asking about flipping coin with same initial conditions and the chosen answer said it's impossible to toss coin and put it again with same initial condition, my question is talking about statistically.

I know that it's impossible to toss coin with same initial conditions according quantum mechanic. But I think it's possible to toss coin with same initial conditions based on classical mechanic relatively.

Now, if we toss coin repeteadly with same initial conditions based on classical mechanic. Will the distribution result still close to 50-50? Or not close to 50-50?

For example tossing it with drop coin from same height with same force and same floor and same initial coin side.

Answer 1

In principle, classical mechanics can predict the outcome of the coin toss provided that we have all the information that must go into the equations of motion. This includes not only height, angle and velocity of the initial condition but also:

• the precise nature of the coin itself: is it perfectly symmetric? if not, how is its asymmetry oriented with respect to the initial impact?

• the precise nature of where the initial force hits the coin (at the center? off center? at a single point? distributed over an areas? what area? etc);

• the precise nature of the collision physics with the table: is the table smooth? what is the friction between coin and table? how does that friction depend on the point of the coin that first hits the table? etc.

• the precise nature of everything in between: are there drafts in the room? is there a temperature gradient? did the coin catch a fuzz during its spin? etc.

The point being, there are so many factors beyond our control that the notion of repeating the toss with the "same initial conditions" must be understood to mean that we only fix those conditions that we can control and let the rest be whatever they happen to be in each repetition of the experiment.

Answer 2

It depends on all the conditions that affect the result, not just the ones you impose. If your imposed conditions dominate then yes you can prejudice the result. For example, drop (not flip) a coin with heads up from 1 mm above a flat surface and it will always result in heads up. Also using a weighted coin can affect the result of a flip. If conditions other than the ones you impose dominate, the result can be equally heads or tails, for example, for flipping a balanced coin from a reasonable height the randomness of the orientation of the coin on impact dominates and heads or tails is equally likely.

Try a few cases yourself.

Answer 3

In an ordinary coin toss, the probability that the coin will land the way it started is about .51. See various papers by Persi Diaconis and co-authors.