Finite State Automata for recognising consecutive characters
The FSA should have three states: 


$q_a$ represents:  last character was not a b, and no bb yet
$q_b$ represents:  last character was a b, and no bb yet
$q_{bb}$ represents:  there was already a bb


Clearly, you start with $q_a$ and you accept in $q_a$ and $q_b$. Now try to figure out how to connect the states.