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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