Context: Many of us know the game War in which players divide a deck of cards to two people randomly. Wikipedia describes the game best (Assume no Jokers, just 2 through A)

The deck is divided evenly among the two players, giving each a face-down stack. In unison, each player reveals the top card on his stack (a "battle"), and the player with the higher card takes both the cards played and moves them to the bottom of his stack. If the two cards played are of equal value, each player lays down three face-down cards and a fourth card face-up (a "war"), and the higher-valued card wins all of the cards on the table, which are then added to the bottom of the player's stack. In the case of another tie, the war process is repeated until there is no tie. A player wins by collecting all the cards. If a player runs out of cards while dealing the face-down cards of a war, he may play the last card in his deck as his face-up card and still have a chance to stay in the game.

So I wanted to create a different game to spice things up because War can get really boring as players know. So I designed (might have been done before) a variation of the game which I called Civil War*. The idea was to maintain the same exact rules except instead of having equal cards, giving one side four aces to begin with. The idea of winning for the side of no aces is to hope for a war and grab aces from the person with the aces.

Question: What is the probability that the person with four aces would win, collect all the cards in the end?

Attempt: The only thing I thought to do is to create a tree diagram in which each divergence is every card challenge. However, at some level the sheer amount of possibilities and minor occurrences to take into consideration made it unrealistic. Even retrying this method with using less cards (only face) for a baseline was impossible. At this point, my brain is fried from looking at probabilities. I don't know how to program, so I don't even have a number. So far I have spent over a week on this with no progress and it is frustrating.

I would appreciate any help of direction, but I am looking for more of an answer than a hint.

*Not political, just thought the name was more fun than War 2 or something.

A: Wikipedia Link

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    $\begingroup$ I don't think you can reasonably make a game tree. It would not be hard to write a simulation and play a bunch of games. $\endgroup$ – Ross Millikan Jan 2 '18 at 3:41
  • $\begingroup$ According to this analysis by students, if a player starts with four kings (highest rank in their version), then he or she has a roughly 85% chance to win the game. $\endgroup$ – vadim123 Jan 2 '18 at 3:46

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