Tradermath logoBrainteaser - Fighting Fluffy

Fighting Fluffy
In an alternate scenario of "Harry Potter and the Philosopher's Stone," Harry Potter faces a unique challenge against Fluffy, the three-headed dog, in the Hogwarts dungeons. Without the flute that usually puts Fluffy to sleep, Harry has to resort to a more direct approach: he must chop off all of Fluffy's heads to defeat him.

However, Fluffy possesses a magical ability to regenerate heads. When Harry attacks, one of three events can occur:

1. Harry successfully chops off \(2\) of Fluffy's heads.
2. Harry chops off \(1\) head, but it immediately grows back.
3. Harry misses entirely, leading to Fluffy growing an additional head.

The probabilities of Harry chopping off \(2\) heads and missing are equal. These events are only possible if Fluffy has \(2\) or more heads. If Fluffy ever grows to \(5\) or more heads, the situation becomes too perilous, and Harry and his friends must retreat. Under these conditions, what is the probability that Harry successfully defeats Fluffy by chopping off all its heads?