If you look at the link I provided, you can see that time is both in the numerator and in the denominator, so it cancels out.
Also, I assumed that the birds are constantly moving around at 10 m/s, which is roughly the flying speed of a bird. And, since I think quantifying bird density to a higher order is very park/environment-dependent, we have to use a more rough estimation. Like I said, my bird density figure might be a little high, but even if you reduce it by a significant factor, there's still a pretty significant chance it will happen in a Major League Baseball game.
Basically I converted the original problem in this: How far does a baseball have to travel before it hits one bird? Since we know how far a baseball travels per pitch
, we can determine the number of pitches this is equivalent to.