Listening to a Parrot

Let $p$ denote the expected number of words uttered before the parrot utters five identical words consecutively. The first word she utter can be any word so let $n=p-1$ . If the parrot mutters the wrong word, then the expected value will just be $n+$ the number of previous amount of words uttered. Due to this, we have the equation

$n=\frac { 4 }{ 5 } (n+1)+\frac { 1 }{ 5 } (\frac { 4 }{ 5 } (n+2)+\frac { 1 }{ 5 } (\frac { 4 }{ 5 } (n+3)+\frac { 1 }{ 5 } (\frac { 4 }{ 5 } (n+4)+\frac { 4 }{ 5 } )))$

Solving for $n$

$625n=624n+500+200+60+16+4$

$n=780$

Therefore, $p=781$