Each problem counts 25 points, including the bonus problem 9, except that problem 6 contains a bonus part worth an additional 10 quiz points. Each part of each problem is weighted equally, except where noted otherwise. You must show all work to receive full credit

1. Let A = {1,2,3,4,5} and B = {3, 4, 5, 6} be events in the sample space S = {1, 2, 3, 4, 5, 6, 7, 8}. Assume all the elements of S are equally probable. Compute each of the following as fractions in lowest terms.
   1. Pr(A)
   2. Pr(B)
   3. Pr(A B)
   4. Pr(A | B)
   5. Pr(A B |A B)
   
   

2. Using the sample space S and the events A and B from problem 1, along with C = {6,8}, determine whether each of the following pairs of events is independent, disjoint, or neither.
   1. A and B
   2. B and C
   3. A and C
   
   

3. The student aviation club at a certain university has agreed to move seven airplanes from one airport to another for their owner. The club has eighteen licensed pilots, any of whom can fly any of the airplanes. Answer the following but do not simplify.
   1. How many different ways can seven pilots be chosen for the planes, if it does not matter who gets which airplane?
   2. In how many different ways can seven pilots be chosen for the planes if it does matter who gets which airplane since they are all of different types?
   3. In how many ways can both a pilot and a copilot be chosen for each plane if it matters who is pilot and who is copilot and which airplane they get?
   
   

4. Two fair dice are rolled. A success means that you roll a sum of 8; anything else is a failure.
   1. What is the probability, p of success?
   2. This experiment is performed 5580 times. Find the expected value of the number of successes.
   3. Find the variance and standard deviation of the number of successes (Use fractions not decimals; the numbers are chosen to come out nicely.)
   
   

5. You have three employees, Harry, Judy, and Kim. If each work two hours, the time billed comes to $100. If Harry works three hours; Judy, one hour; and Kim, two hours, the time billed comes to $99. If Judy and Kim each work three hours, the cost is $108. Using a system of linear equations, solved by matrix row operations, find each persons hourly charge (i.e., pay rate).

   
   

6. Assume w, z, y, z 0 and that x +2y +3z + 4w=36.
   1. Find the maximum possible value of P=8 w-x using the big-M version of the simplex method discussed in class. (You will need an artificial variable but no surplus or slack variables.)

   2. Find the minimum value of the same function subject to the same restriction by the same method. (Hint: Pivot on the x-column fist, after the setup.)
   
   

7. Set up the initial tableau, with appropriate slack variables, etc. for the following problem. Then identify the first pivot, showing your work. Please do not solve the problem completely.
   
   A firm produces two kinds of turkey stuffing during the holiday season, regular and special sage. Each box of regular mix requires 12 ounces of bread, 4 ounces of cornbread-sage mix, and 1/2 ounce of spices. Each box of special sage mix requires 5 ounces of bread, 3 ounces of cornbread-sage mix, and 1 ounce of spices. The profit on one box of regular mix is 45 cents, and the profit on one box of special sage is 30 cents. Each day the firm has 3,360 ounces of bread, 960 ounces of cornbread-sage mix, and 160 ounces of spices. Determine the number of boxes of each kind of stuffing mix that should be produced each day to yield maximum profit.
   

8. A Markov chain with two states is sometimes referred to as a system of Markov-dependent Bernoulli trials. If the two states are success (S) and failure (F), and the following transition matrix describes the transition probabilities, find the fixed probability vector. What is the long run probability of a success?
   
   Transition Matrix:
   
   (Note: The answers will be in terms of fractions with denominator equal to eleven. Full credit will not be given if you use decimal approximations.)
   

9. Bonus: 25 quiz points

   You have five urns numbered 1 through 5. Each urn contains two numbered balls as indicated in the following chart:

   urn	1	2	3	4	5
   Balls	1,1	2,2	1,4	2,5	3,5

   
   A ball is drawn randomly from urn 5. Thereafter, a ball is drawn randomly from the urn numbered the same as the previous ball. Each ball drawn is put back into the urn it came from after its number is recorded. In the following parts d) and e) count 5 points each; the other parts are three points each.
   
   
   1. Write down the 5 x 5 transition matrix fro this Markov chain.
   2. How many absorbing states and how many transient states are there?
   3. Write down the matrices Q and R for this Markov chain?
   4. Compute (using fractions) N=(I-Q)^-1
   5. Compute C=NR
   6. What is the probability this process ends up stuck in urn 2?
   7. If you start in urn 4 instead of 5 what is the probability you end up in urn 2?