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))

then

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.