There is no mathematical basis for this general statement, however,
there have been cases where people caught on to the sequence coming
from a poorly- or non-reseeded pseudo-random number generator
in a casino game and won hundreds of $thousands before the
casino realized their error...
The most important factor in gambling is the size of the house.
There is also no mathematical basis for THIS statement...you
guys are batting .000 again...
In a fair game, the player with the biggest house money volume wins.
Oh, a "fair game"...who the hell offers a "fair game"? In any event,
it's irrelevant, because the actual most important factor (to the extent
that we indulge in the pointless semantics of pronouncing a "most
important factor") is the "expectation" of the game.
Assuming a Markov process (random walk) the players will get ahead and
behind in a random, wobbling fashion. But as soon as the cash for one
player is gone, the game is over. If you have one hundred dollars and
the opponent has one trillion dollars, you are in a lot of trouble.
If you have a casino, and are stupidly offering a "fair game"
(0% "expectation"), you will eventually lose all your $trillion
to salaries and other expenses no matter how many individual
players come in and lose $100, because the money you win
from them will be offset by players who "get lucky" and win
$100, $200, $3000, or more...
You would be correct if you asserted that there is relationship
between the size of your "bankroll" and your average bet size
as a fraction of that "bankroll" in terms of actually acheiving
a result of "bankroll" growth (or non-loss) that conforms to
your "expectation" for the game. But that's a slightly more
complicated concept, innit?
That's why I call gambling "A tax on stupidity."
Stupidity is kind of its own tax, innit?