Consider a binary message in which $0$ has has probability $1/3$ and $1$ has probability $2/3$. What value of $H$ should be assign?
I know that you split up $1$ into two messages $1a$ and $1b$. Then I think you have to use conditional probability here. Am I on the right track? So $$\left[\text{entropy of} \ (0,1a, 1b) \ \text{message} \right] = \left[ \text{entropy of} \ (0,1) \ \text{message} \right] + \text{something}$$