I’m preparing for a final and I’ve stumbled upon the following problems that I just can’t get right. Any help with them will be extremely appreciated!

Markovian chain problems

1.A carnival man moves a pea among three shelves, A, B and C. Whenever the pea is under A, he moves it with equal probability to A or B( but not C) When it is under B, he is sure to move it to C. When the pea is under C, it is equally Likely to move it to A, B or C.

1. Set up a three state Markov chain by drawing the transition diagram and check whether it is regular

2. Find the transition matrix and evaluate the probabilities for the pea to be under each shell after two moves, if it is initially under C

3. Find the long range probabilities for the pea to be under each shell

4. Find the probabilities for the pea to be under each shell after two moves, if you know that it was initially under C and the next time it is not under B

2.A professor tries not to be late for class too often. If he is late some day, he is 90% sure to be on time the next time. If he is on time, the next day there is a 25% chance of his being late. In the long run, how often is he late for class?

Linear programming

A farmer owns a 200-acre farm and can plant any combination of two crops 1 and 2. Crop 1 requires 1 man-day of labor and $10 of capital per each acre planted, while crop 2 requires 4 man-days of labor and $20 capital per acre. The farmer has $2200 of capital and 320-man-days of labor available for the year. Crop 1 produces $40 of net revenue and Crop 2 produces $60.

a) Find the optimal strategy

b) Because of some tricks on the stock-market the price of Crop 2 begins growing up. What is the turning point of the price of Crop 2, when the farmer should recalculate the maximum revenue scheme?

Probabilities

1.Find the probability for the following poker hands:

a) Full house which contains at most 2 Face cards

b) Three of a kind, given that it contains exactly one club

c) Four of a kind, given that is contains at least 3 red cards

2.An urn contains 2 Black and 4 Red balls. A sample of 2 balls is drawn and then withought replacement two more balls are pulled out

a) Check whether the events:

P: The balls of the first sample have the same colour

And

Q: The balls of the second draw have the same colour

Are independent?

b)Find the probabilities for each of the following:

A: The balls of the first draw have the same color, given that both balls of the second sample are red

B: The balls of the first draw have different color, given that the balls of the second one have different colour.