Multiple Criteria, non-binary outcomes

So, I'm trying to predict outcomes using baseball data. What I want to do is predict the outcome of batted balls based on the following criteria:

Pitcher: 50% groundballs allowed, 20% line drives allowed, 25% flyballs allowed, 5% popups allowed.
Batter: 40% groundballs hit, 15% line drives hit, 35% flyballs hit, 10% popups hit.

I want to predict the outcome of these two matching up. To do that, I think you also may need to know the league averages. Let's just say:

Lg avg: 45% GB, 20% LD, 30% FB, 5% POP.

So in the past I'd used the odds ratio method. Like if using the above, we just wanted to figure the odds of 'line drive' versus 'other', I believe we'd go about that by saying:

(LD ratio) = (hitter LD rate / (1-hitter ld rate)) * (pitcher LD rate / (1-pitcher ld rate)) / (league LD rate / (1-league ld rate))


Odds LD = (LD ratio)/(LD ratio +1)

But I think this only works for binary outcomes, right? "Line drive" versus "other". If I want to calculate the odds of the above matchup with respect to all the batted ball types, what's my methodology for getting there?

Thanks in advance.
Let P=p be the event that pitcher's throw results in event p (GB, LD, etc.), and similarly B=b be the event that batter hits a "b"

Are P and B independent? If so, then Pr(P=GB & B=GB) = .5*.4 = 20%, for example
Sorry! I should have been more clear about that.

But forgive me if I'm not clear what the correct answer is. I can give you the relevant details...

The individual instances are indeed independent, in the sense that the outcome in one trial is in no way chained to the next.

But the outcomes in the first place can be thought of as 'buckets'--different gradients of the same thing. I don't know if that makes them 'dependent' in any way. Whether you're familiar with baseball or not, I hope this makes enough sense to be useful--a ball is struck; the angle and velocity at which it is struck will determine which of these buckets it falls into. So what we're ultimately trying to do here is... pitcher A has the batted ball tendencies of x%GB/y%LD/n%FB/q%POP, hitter has a different distribution of the same categories, (and the league has norms), so based on this, what should we expect from the resulting matchup?

Thank you again and I hope this makes things clearer.