discrete mathematics - How to express this multi-player card probability mathematically?

I am having trouble conceptualizing how this would be expressed mathematically and could use your help! I have access to Maple and MATLAB so feel free to post examples.

The Problem:

Bill and John are playing a game where they each draw cards from a deck of 52. If they draw an Ace of Hearts, they win 100 dollars. If they draw a King of Hearts, they win 50 dollars. If they draw any other card, they lose 1 dollar. They can draw as few or as many cards as they want (essentially drawing without replacement).

The Ask:

What is the probability of Bill selecting an Ace of Hearts out of his 5 next cards, given that he has already lost 12 times and there are 30 cards left (John has played and lost also)? Remember, as each player draws more cards the probability of the next card being an Ace of Hearts increases.

Optional Ask:

Calculate Bill's expected value for playing this game, given that John has already played 10 cards and lost. Right now I am getting stuck on this calculation when scaling to multiple prizes; here is what I have so far:

$$\displaystyle{a}=\frac{{\Gamma{\left({M}+{1}\right)}\cdot\Gamma{\left({N}-{M}+{1}\right)}}}{{\Gamma{\left({M}-{K}+{1}\right)}\cdot\Gamma{\left({N}-{M}-{P}+{K}+{1}\right)}}}$$

which equals the number of arrangements for each prize. K is the number of prizes to select, P is the number of prizes in the game, M is the number of cards Bill chooses, and N is the number of cards left in the game. (Getting a feeling of hypergeometric distribution here)

and
$$\displaystyle{c}=\frac{{\Gamma{\left({N}+{1}\right)}}}{{\Gamma{\left({N}-{P}+{1}\right)}}}$$
for the number of total cases.

How do I add the prizes into the mix to calculate expected value?

Notes:

The formula(s) must be scalable to more than 2 players, more than 2 prizes, and more than 52 cards.

Thanks!