How to determine the randomness of coin tosses by its final position? Suppose a circle is drawn on the floor, and a coin is tossed over it many times. If it jumped out of the circle, then we do not take it into account. How to determine from the statistics of coin end locations, that the game is really random and there are no fraud?
It's clear that the distribution of coins on the floor should tend to be uniform. But how to numerically determine the probability of randomness by the resulting statistic and number of throws?
 A: Unless the circle is small relative to the areas in which the coin is landing, there is no reason to believe that the distribution of coin will be uniform over the circle.  Indeed, if you were to examine the landing point of the coin without conditioning on it being in the circle then it would certainly not be uniform.  Now, if you take a small enough circle relative to the areas where the coin is landing (so that there is low probability that the coin lands in the circle) then the conditional distribution on the circle could reasonably be approximated as a uniform distribution (on the basis that the true density is likely to be continuous and changing only slowly within the circle).  That is only a reasonable assumption if the circle is small.
Now, if you use this setup, there are lots of ways to test for uniformity of the data.  Many tests for uniformity over a continuous space use "binning" to divide the space into equal-sized discrete parts and then they test for discrete uniformity.  Common tests for discrete uniformity are the chi-squared test (when you have a lot of data) or occupancy test (for sparse data).
