MLE for given HMM and Observation Sequence

Question

We are given an MDP with 3 states: Red, Green, Blue (RGB). 

Emission probabilities:  

R --> 1 (p=1/3), 2 (p=1/3), 3 (p=1/3)

G --> 1 (p=1/3), 2 (p=1/3), 3 (p=1/3)

B --> 1 (p=1/3), 2 (p=1/3), 3 (p=1/3)

Transition Matrix

R --> R (p=1/3), G (p=1/3), B (p=1/3)

G --> R (p=1/3), G (p=1/3), B (p=1/3)

B --> R (p=1/3), G (p=1/3), B (p=1/3)

These are the observations: 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2

What is the most likely explanation for this observation sequence?
Initial state is not known and is equally likely from all states.

Answer 1

Without solving this problem mathematically, I think, it is very obvious that the most likely explanation for this observation sequence is about their equality as emission probabilities and values of transition matrices are the same. Therefore, there is no unique MLE; any state sequence of the same length is equally probable. 

If we solve this problem using the mathematical equations:

P(S,O)=P(s1​) ∏( P(st​ ∣ st−1​) ∏ ​P(ot​ ∣ st​)

P(s1​)=1/3

​P(st∣st−1)= 1/3

​ P(ot∣st)=1/3​

After putting the values into the equation, we get :

P(S, O)=(1/3​) ^ 2T, which means P(S, O) is independent of S, so whichever state sequence is there, they all have the same likelihood.