I am looking for a generalized answer to a problem like the following.
What is the probability of drawing S spades before, D diamonds or C clubs or H hearts.
The purpose of this is to be able to determine the win probability for a given suit in the following game.
Remove all 4 aces from the deck of cards and place the horizontally on the table, then draw the top X cards from the deck and align them vertically ( where X+1 will represent the game length). Below is an example where X = 1, and the card drawn from the top of the deck was the 7 of hearts (7H).
7H
AH AD AC AS
From here to win the game cards are drawn from the top of the deck and the Ace that corresponds to the suit drawn gets to move forward, when a specific suit gets to the end the game is over.
So for the scenario above if we want to know the probability of diamonds winning we would need to know the probability of drawing 2 diamonds before drawing 2 hearts, or 2 spades, or 2 clubs.
To simplify the game further I started with the base case of X = 0.
AH AD AC AS
In this scenario the chances of any suite winning would be 25% (12/48)
From here I tried to work out the case for X=1 but other than manually calculating all of the possible scenarios I wasnt able to see how the problem can be generalized for any value of X.
any help would be greatly appreciated.