[Show all your work, including all row operations. The correct answer with no work will be worth only half credit. Problems 1-8 count 25 points each; problem 9 is worth 20 bonus quiz points. Problems 8 and 9 are to be done by the simplex and/or big M method. Only 60% credit will be given for a correct graphical solution for 8 or 9. (i.e. 15 points for # 8, 12 points for #9.)]

1. A certain surgical procedure has a probability of success of 9/10. Over a period of months, the procedure is performed 100 times, making 100 Bernoulli trials. Find the expected value, variance, and standard deviation of the number of successes.
   

2. You have three urns, numbered 1, 2 and 3. Urn 1 contains one red ball and one green ball; Urn 2 contains two red and two green; and Urn 3 contains three red and three green. You chose an urn randomly and draw two balls without replacement. Given that the two balls drawn were of different colors, find the conditional probability that Urn 1 was selected.
   

3. The transition diagram here is taken from a problem in your textbook:
   
   Write down the transition matrix P.
   1. Compute P².
   2. Explain what the entries in P² mean.
   3. Find the fixed probability vector.
   
   

4. [You need not do any arithmetic for this problem.] A hand of five cards is dealt randomly from a deck. Let A₀, A₁, A₂, A₃, and A₄ denote the events: "no aces", "exactly one ace", "exactly two aces'" etc.
   1. Write down the probability of each of these events.
   2. Write down the expected value of the number of aces dealt.
   
   

5. Let A = and B =
   If AB = , find both a and b.
   

6. One of the following matrices has an inverse, the other does not. Find the inverse of the one which does and explain why the other does not.
   1. 
   2. 
   
   

7. Ted's Toys makes toy airplanes, boats and cars. The materials used are plastic, wood strips, and steel. Each airplane uses 100 grams of plastic, 10 inches of wood strips, and 80 grams of steel. Each boat uses 50 grams of plastic, 9 inches of wood strips and 50 grams of steel; and each car uses 70 grams of plastic and 120 grams of steel. If Ted's has on hand 12100 grams of plastic, 5220 inches of wood strips, and 10800 grams of steel, how many planes, boats, and cars should be made to use up all these supplies prior to shutting down for retooling?
   

   [see comments at the beginning of test concerning problems 8 and 9]
   

8. Given that x, y 0 and 2x + y 50 and 2x + 3y 118, find the maximum possible value of x.
   

9. 20 quiz points

   You discover that the situation in problem 8 was misrepresented, and that and 2x + y 50 should have been and 2x + y 50. Find the maximum possible value of x in this case.