In the abstract, they operate by randomly selecting a fixed set of numbers from a larger pool.  For instance, they might select 6 numbers from a pool of 42 (as was the case in Bulgaria).  The exact method is irrelevant; they might use numbered ping-pong balls, perhaps a computer random-number generator or any other method; as long as it can be guaranteed to be random.

This immediately raises some confining rules.

Firstly, that once selected, a number cannot be re-selected.  In other words, duplicates are not allowed.

Secondly, order is not important.  Choosing 1, 2, 3, 4, 5, 6 produces exactly the same result as 6, 5, 4, 3, 2, 1 (or any other combination). In fact, the Bulgarian lottery did produce the numbers in different order.

So, what are the odds of any specific set of numbers?  Let's calculate the odds of winning the Bulgarian lottery (based on drawing numbered balls from a bucket).  Being limited by the two confining rules, we know that the odds of a player choosing the first ball are 1 in 42.  However, for the second ball the odds are only 1 in 41 (because a ball has already been removed from the pool).  Subsequently, it reduces with each ball drawn out.

So, in order to match the balls as they are drawn, our player has odds of 1:42 x 1:41 x 1:40 etc down to 1:37.  This is a huge number - approximately 1:3.78 billion.  On that basis, no-one would ever win!

However, we have not constrained ourselves by the second rule.