Probability of getting a bingo pattern with a certain number of balls drawn?

Let's say you have a fixed pattern on a standard 5x5 bingo card using all the standard bingo rules (center is a free space, B column has numbers 1-15, I has 16-30, etc. to 75). How can you calculate the probability of matching at least that pattern in X balls drawn? You can have other spots marked too.

Specifically I'm looking for the odds of hitting the hot dog pattern in 17 ball draws, but I'd like to know the method in order to generalize this.

Does the probability depend on the number of cards in play?

Thanks!

There are $10$ non-free squares in your pattern. There are $\binom{65}{7}$ ways to draw the remaining $7$ balls from the remaining $65$ balls, and $\binom{75}{17}$ ways in total to draw $17$ balls from $75$ balls, so the probability is
$$\frac{\binom{65}7}{\binom{75}{17}}=\frac{65!\cdot17!}{75!\cdot7!}=\frac{104}{4432786665}\approx2.3\cdot10^{-8}\;.$$