> Maybe Don can help here, but it might also
> be true that some of these formulas break
> down a bit when given extreme parameters
> (like 1 hour of play only).

That is true:
The mathematical formulas assume the so-called Brownian motion , which is a continuous version of the so-called random walk. For a play of 1 hour the approximation is not bad since 1 hour results can already be approximated by normal distribution with not too much error.

However, a large error could be caused by "overshooting" the goal. In other words, the winning goal should be large relative to the maximum bet size, for example.

The formula is grossly wrong in the example above, with $1 winning goal. The actual probability that the player walks away with at least $1 is smaller since in most cases he would win signification more than $1.