I am learning Hidden Markov Models and trying to solve this problem. A Hidden Markov Model has 2 states and 3 observations . The state transition matrix (A), observation matrix (B)and initial matrix ( ) of the model is given below.

(a) if the observation sequence is and the hidden sequence is , calculate the probablity that the next observation is

(b) For question (a), if the next observation is b, what is the probability that the hidden sequence is

Now I have some questions here. For part (a), using the conditional probability laws, I would have

Is this correct ?