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

[answerhappygod]

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)
= 4/8 P(amount of rain = 2 thousand hm3) = 1/8 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.
(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 Monday, find the
probability that the water level in the dam will be at 2 thousand
hm3 on Thursday of the same week. (8 pts) (Four decimals)