How to evaluate Markov Model accuracy

Question

I have created the following Markov chain Model. And I am struggling to prove mathematically that my model works correctly, or doesn't work.

Sequence: Start, state1, state2, state3, state3, state2, state1, state2, state1, end

States: start, state1, state2, state3, end

Distribution:

Start: 1
state1: 3
state2: 3
state3:2
end: 1

Pairs of Token:

(Start,state1):1
(state1, state2):2, (state1,end):1
(state2,state3):1, (state2, state1): 2
(state3,state3):1, (state3, state2):1
(end, ‘None’):1

Possible tokens to follow each key:

Start:  [state1]
state1:  [state2, end]
state3: [state3, state2]
state2: [state3, state1]
end: ‘None’

Transition Matrix:

         Start   state1    state2   state3    end   
Start     0         1        0         0       0  
state1    0         0        0      0.666...  0.33...
state3    0         0        0.5      0.5      0
state2    0        0.66...   0.33...   0       0
end       0         0        0         0       0

I use MLE to calculate transition matrix.

Question: How can I prove mathematically, if my Model is working… e.g. like calculating the mean error squared.

I have an Idea. If I take the sequence as the right one. And try to make suggestions using the matrix. In each step, I will test if the suggestion is the same comparing to my proven sequence. And so sums up the error (error_sum). My Error will be error = error_sum/ all_steps.

Theoretically, this will work. But what I am looking for is a method which is mathematically proven, and I can demonstrate why this method was a good idea to use. Can you provide me, with any suggestions?