Hi first divide the given time period into number of moves
3 hours / 45 mins = 180 mins / 45 mins = 4 moves
Then draw the transition matrix (Since the probabilities are given for each of the could be go up , go down or stay put)
The transition matrix ( P )will look as follows
The distribution over states can be written as a stochastic row vector x with the relation x(n + 1) = x(n)P.
So if at time n the system is in standard, then four time periods later, at timen + 4 the distribution is
So after four states it will be x(n+4) = X(n) P ^ 4
X(n) = [ 1 0 0 0] (where 1 represents Standard )
This X(n) table must be multiplied with P ^ 4
[1 0 0 0] [ 0.477 0.053 0.390 0.081
0.459 0.055 0.403 0.083
0.460 0.053 0.400 0.088
0.420 0.039 0.401 0.140 ]
= [ 0.477 0.053 0.390 0.081 ] == [ standard Genius Acceptable Fail]
Which means the likely state is Standard with probability 0.477
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