theory of uneven distribution

In small samples of independent events, it is unlikely to get perfectly even distribution. This can be demonstrated in many ways:

Roll an N sided dice N times.

Roll a 6 sided dice once.

Roll it again (#2). The chance of rolling a number not yet rolled is 5/6.

Roll it again (#3). The chance of rolling a number not yet rolled is 4/6 (2/3).

Roll it again (#4). The chance of rolling a number not yet rolled is 3/6 (1/2).

Roll it again (#5). The chance of rolling a number not yet rolled is 2/6 (1/3)

Roll it again (#6). The chance of rolling the remaining number that has not been rolled is 1/6.
Thus, for rolls of a 6 sided dice, the chance of rolling 6 unique numbers is the product (and) of each of the above probabilities:
1 * 5/6 * 4/6 * 3/6 * 2/6 * 1/6
This is 1.54% that on 6 rolls of the dice, that each number will appear exactly once.

Flip 4 coins

It is expected to get 2 of them to be heads, and two of them to be tails. There are 2⁴ (16) ways to flip these four coins:

T T T T : 1

H T T T : 4

H H T T : 6

H H H T : 4

H H H H : 1
While it is true that 2 heads and 2 tails is the most probable single selection in that list, it is also the case that 2/3 of the combinations are not the most likely.

It is true that over time, these events will average out. Approximately 1/6th of the dice rolls will be any given number, and half of the coin flips will be heads. This is not the case for small samples of these events though.