MATH 230

Final Exam

Fall 1999

Bellamy & Williford

Except as noted for problems 3,4,5 and 8c), each problem counts 25 points. Show all your work.

1. 25 points

   Let A ={1,2,3,4,5,6,7} Determine how many elements are in each of the following sets.
   1. 
   2. 
   3. 
   4. are not all the same}
   5. are all different}

2. 25 points

   You have 20 books on a shelf. You choose 3 randomly to read on a trip. Two of the twenty have $100 bills in them, which someone used as bookmarks. Let X= the number of books you chose with $100 bills inside. Compute:
   1. Pr(X=0)
   2. Pr(X=1)
   3. Pr(X=2)
   4. E(X)
      (You do not need to do any arithmetic.)

3. 15 points

   1. Evaluate P(10,4)
   2. Evaluate C(10,4)
   3. Explain in your own words why P(10,4) > C(10,4) without reference to the formulas, just the definitions.

4. 20 points

   Two fair dice are rolled, one red and one green. Find the conditional probability that the sum is not six given that it is not seven.

5. 15 points

   1. Perform whatever row operations are necessary to put this matrix into reduced form.
      
      

   2. Find the inverse of: showing each row operation.

6. 25 points

   Using the simplex method, solve the following maximum problem: (Half credit for the correct answer by any other method.)
   
   

7. 25 points

   A regular Markov chain has the following transition diagram:
   
   1. Write down the transition matrix for this Markov chain.
   2. If the process starts in state II, what is the probability it is in state II after two steps?
   3. Find the fixed probability vector for this Markov chain.

8. 25 points

   A random variable X has the possible values 1, 3, and 5. Pr(X=1) = p; Pr(X=3) = q; Pr(X=5) = r. Given that E(X) 3, find the largest possible value of E(X²). (Recall: p + q + r = 1; E(X) = p + 3q + 5r; and E(X²) = p + 9q + 25r.)
   1. Set this problem up using the big M version of the simplex method.
   2. Perform the necessary operations to get the tableau in proper initial form.

   3. Bonus: 15 quiz points

      Solve the problem, showing all your row operations.

9. 25 points

   Compute all the following which are possible to compute; explain why for any impossible ones:
   
   
   

10. 25 Bonus quiz points

    The following transition matrix models an absorbing Markov chain which can be set up using coin tosses. If it begins in state 3, what is the probability it ends up in state 1?