In the online card game, Hearthstone, there is a play mode called Arena. Players who enter the arena build decks (commonly referred to as drafting) and play until they win 12 matches or lose 3 matches. Since the decks are built 'randomly,' and the matches are also random, we can assume that each player has a 0.5 chance of winning and a 0.5 chance of losing.
What would be the best way to calculate the probability of ending with exactly 0 wins, 1 win, 2 wins, ..., and 12 wins? Since the arena ends after any 3 loses there are branches of the binomial tree that would never occur.
ie Completing 3 matches have the following possible outcomes. WWW, WWL, WLW, WLL, LLL, LLW, LWL, LWW. Since LLL fulfills the 'lose 3 match requirement' LLLL and LLLW can not occur. Likewise during a 5th round, WLLLW, WLLLL, LWLLW, LWLLL, LLWLW, LLWLL, LLLWW, LLLWL, LLLLWL, LLLLL can not occur.