In the trading card game Magic: The Gathering, there is the opportunity for combos to extend infinitely. Since some cards involve probabilistic events, it is possible for a combo to be nondeterministic in outcome. For example, in this video by FishMTG, a situation is contrived in which the opponent has 4 life and controls only a tapped Leonin Elder, and the active player player can activate a Wirefly Hive as many times as they want. The effect of this is that a coin is flipped repeatedly:
- On heads, the active player gets a $2$ damage attacker and the opponent gains $1$ health.
- On tails, the active player loses all of their attackers.
If the active player gets attacking damage equal or greater than the opponent's health, they can attack and win the game. The question is how probable is it for the active player to win? Even though they infinite opportunities to get the flips necessary, the longer they go, the more heads in a row they need. Precisely, they need a run of heads that is greater than or equal to half of (the total number of heads$+4)$
Coding a simulation in python, I get an approximate answer of $0.183$ (with a max health of 10000 because the chance of getting 10000 heads in a row is almost impossible). You can read my (very bad) code here. But I'd like to know how to calculate a more precise answer.