Brainteaser iconBrainteaser - Fighting Fluffy

Fighting Fluffy
Probability
Citadel, Optiver
In an alternate scenario of Harry Potter and the Philosopher's Stone, Harry Potter faces a daunting challenge in the Hogwarts dungeons: defeating Fluffy, the three-headed dog that guards the trapdoor to the Philosopher’s Stone. Without the magical flute that usually puts Fluffy to sleep, Harry must rely on brute force to chop off all of Fluffy’s heads to defeat him.

Fluffy begins with \(3\) heads, and during each of Harry’s attacks, one of the following can happen:

1. Harry chops off \(1\) of Fluffy’s heads, but it immediately grows back.
2. Harry misses, and Fluffy grows an additional head.
3. Harry successfully chops off \(1\) of Fluffy’s heads.

The probabilities of chopping off \(1\) head and Fluffy growing an additional head are equal, while the probability of the first outcome (chopping off \(1\) head and it regrowing immediately) is distinct.

If Fluffy’s head count ever reaches \(0\), Harry wins. If it grows to \(5\) or more heads, the situation becomes too perilous, and Harry must retreat. What is the probability that Harry defeats Fluffy?