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.

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.

http://www.experts-exchange.com/Other/Math_Science/Q_28211392.html

Second to last post.

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 much larger numbers of rounds use a transition matrix as ozo described in this question

http://www.experts-exchange.com/Other/Math_Science/Q_28228054.html

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.

ExEx-Probability-3b.bmp

ExEx-Probability-3d.bmp

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if the first statue inspected is Superiorthen the probability that the thirdstatue 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