Still not as hard as Dark Souls

Discrete Mathematics Level 4

The video game Schmark Schmouls is an excellent video game with the latest graphics and cutting-edge gameplay and will be a surefire hit when it releases sometime in the near future (Soon™). Also, Schmark Schmouls has no relation to Dark Souls... please don't sue.

In the game, the player starts at checkpoint \(A\), and must pass through the checkpoints \(B\), \(C\), and \(D\) in order. After passing checkpoint \(D\), the player wins the game. To pass each checkpoint (except for the starting checkpoint \(A\)), the player must defeat a boss.

Each time a player attempts to defeat a boss, the following outcomes could occur:

  • \(\frac{1}{3}\) probability: The player defeats the boss and passes the checkpoint.
  • \(\frac{1}{3}\) probability: The player loses to the boss and gets sent back to any of the previous checkpoints with equal probability. The boss at that checkpoint will not re-spawn, but all bosses in succeeding checkpoints will re-spawn.
  • \(\frac{1}{3}\) probability: The player loses to the boss, throws the controller across the room, turns the game off, cries in the corner, and never plays Schmark Schmouls again.

The probability that a player wins the game of Schmark Schmouls can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are positive co-prime integers. Find \(a+b\).


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