The engineer responsible for the maintenance of a dam is in charge of keeping the water level in it within certain limit

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The engineer responsible for the maintenance of a dam is in
charge of keeping the water level in it within certain limits.
There is a water level known as level 0, from
which certain security measures are applied. For the dam control
purposes, the level of the water is considered each thousand cubic
hectometers (hm3). When the monitoring system is put in
place, the water level is one thousand hm3 above level
0. Each day the engineer monitors the water level and applies the
following policy: if the water level in the dam is grater than or
equal to 3 thousand hm3 above level 0, he lets out
enough water to leave the level of the water at 2 thousand
hm3, if the water level is less than 3 thousand but more
than (or equal to) 2 thousand hm3 above level 0 then he
lets out 1 thousand hm3 of water, if the water level is
less than 2 thousand hm3 above level 0 at the time of
inspection then he does nothing. Keep in mind that only multiples
of thousands are considered for the water level. The probability
distribution of the amount of rain over the dam on any given day is
the following :
P(amount of rain = 0 hm3) = 3/8
P(amount of rain = 1 thousand hm3) = 1/8
P(amount of rain = 2 thousand hm3) = 4/8
Express the situation of the water level control of the dam as a
Markov chain, by defining the variable as xt =
water level in the dam before the engineer lets
the water out according to the policy at period t.
IN YOUR
DOCUMENT ANSWER THE
FOLLOWING:
(a) Write the one-step transition matrix for this problema
(hint: note that state 0
is not posible) (17 pts)
(b) If the monitoring system is put in place on Tuesday, find
the probability that the water level in the dam will be at 2
thousand hm3 on Friday of the same week. (8 pts)
IN THE SPACE
PROVIDED WRITE
THE PROBABILITY
FOR PART (b) TRUNCATED
TO FOUR
DECIMALS: