probability exam question3

I am trying to fit a binomial formula to these 2 questions but I am struggling

Victoria Jones runs a small business making and selling statues of her cousin the adventurer Tasmania Jones.
The statues are made in a mould, then finished (smoothed and then hand-painted using a special gold paint) by
Victoria herself. Victoria sends the statues in order of completion to an inspector, who classifies them as either
‘Superior’ or ‘Regular’, depending on the quality of their finish.
If a statue is Superior then the probability that the next statue completed is Superior is p.
If a statue is Regular then the probability that the next statue completed is Superior is p – 0.2.
On a particular day, Victoria knows that p = 0.9.
On that day

b if the first statue inspected is Superior, find the probability that the next three statues are Superior

On another day, Victoria finds that if the first statue inspected is Superior then the probability that the third
statue is Superior is 0.7.
d. i. Show that the value of p on this day is 0.75.
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Make the same sort of probability tree that I described in one of your earlier questions.

Second to last post.
jagguyAuthor Commented:
this may also be just a tree but what method can you use if the there is a lot of consecutive statues. using a tree wont solve all problems quickly

On this day, a group of 3 consecutive statues is inspected. Victoria knows that the first statue of the 3 statues
is Regular.
ii. Find the expected number of these 3 statues that will be Superior.
For three or four rounds use a tree.

For much larger numbers of rounds use a transition matrix as ozo described in this question
Here are the trees.  Sorry my sketch pad is acting up.

Part b is straight forward.       (0.9)³ = 0.729

Part d is simple but tedious.  You have to find the sum of the four terms corresponding to Superior on the third day.  All of the terms are cubic (of order p³).

     p³  +  p(1-p)(p-0.2) + (1-p)(p-0.2)p + (1-p)(1.2-p)(p-0.2) = 0.7

There is no better way.

>>   if the first statue inspected is Superior then the probability that the third
      statue is Superior is 0.7

The tree for 3d is correct, but you only have to deal with the left hand side.
So the tree is two levels deep (Statue 2 and Statue 3) and the equation you have to solve has two quadratic terms.

         p² + (1-p)(p-0.2)  =  0.7

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