There is a mathematical way to express what you are seeing across these different events. This also relates to two different topics I posted a while back a topic called the Potential Reoccurrence Probability and the Potential Occurrence Probability. Both of these are related, but to show how they are and how this relates to your problem, we'll need to do a little Calculus. The Potential Reoccurrence Probability is the following equation:
y = e -(x / m)
x is the difference between the last draw and the current draw and m is the average rate reoccurrence of these differences. y tells what is the probability that an event will reoccur given its average rate of reoccurrence and the draws since its last event. As the number draws increase without a reoccurrence of the event, there is also another probability that the event will occur. This probability is the measure of the work already done by its potential reoccurrence relative by proportion to its average rate of reoccurrence. Every time an event does not occur, it builds up a potential of occurrence by integrating these values through its draw difference. The basic definite integral looks like this.
y = m-1 [a to b] ò e -(x / m) dx
This then becomes
y = m-1 (-m e -(x / m) ½ [a to b])
The limits of the integral are then a = 0 and b = x. b is equal to x because we are looking for the occurrence for the same draw difference as our original equation. This works out to the following.
y = m-1 (-m e -(x / m) ½ [0 to x])
y = m-1 ((-m e -(x / m)) - (-m e -(0 / m))
y = m-1 (-m e -(x / m) + m e -(0 / m))
y = (m / m) (- e -(x / m) + e -(0 / m))
y = - e -(x / m) + e -(0 / m)
y = e -(0 / m) - e -(x / m)
y = 1 - e -(x / m)
This equation shows the probability an event will occur relative to its last draw and in proportion to the total possible reoccurrences since then to its average rate of reoccurrence. This equation can explain why you are seeing this phenomena of proportion by 6 in most everything. We need to convert this equation into a percentage by multiplying by 100%. Also, when evaluating the equation it's best to round the values to an integer value. In your case, will be looking for when does x by proportion to m is the value of y in percent equal to 100% first and only first to be 100%. In addition, we'll give this proportion an integer value to keep in line with the integer measure of y. The proportion you are looking at is as follows.
x = n · m
n is an integer factor that you are looking at and when n = 6 is the phenomena point you are observing. If we substitute in to the potential occurrence equation we get this.
y = 100% · (1 - e -((n · m) / m))
This reduces down...
y = 100% · (1 - e -n)
This is the general equation that fits for any phenomena because the average rate of reoccurrence factored out and is now relative to your observations in terms of n. n is the number of times the phenomena's average rate of reoccurrence has happened. Now we can apply it to some values and see what happens for any kind of phenomena. Below is a table showing the percentage of total occurrences that have happen since the last draw.
As you can see, by time we get to the 6th n value, there will be in every phenomena an integer percentage of 100% occurrence. This is exactly what you are seeing, the magic #6.