MATH 230

FINAL EXAMINATION

FALL 1998

Dr. Bellamy & R. Sullivan

Each Probelm counts 20 points. In addtion, 7iii) is worth 10 Bonus quiz points.

1. A market researcher interviews three students chosen randomly from a class of 10. Four of the students in the class own large-screen television sets. What is the probability that none of the people interviewed owns one of these TV sets? (You need not simplify your answer.)
   
   

2. Let A,B and be events in a smaple space satisfying the following conditions: and are independent; ; and Suppose . Find . and . (A Venn diagram may be helpful. Remember the difference between independent and disjoint!) Express your answers as fractions in lowest terms.
   
   

3. You have an urn which contains three fair coins and one two-headed coin. You choose a coin randomly and toss it twice. Both tosses yield heads. What is the conditional probability that you chose the two headed coin from the urn?
   
   

4. You have 12 eggs, two of which are rotten. You choose 4 of them randomly. Write down the expected value of the number of rotten eggs among these four. (Leave your answer in combination or permutation form if you wish.)
   
   

5. A fair coin is tossed one million times. Find a) The mean, or expected, number of heads. b) The variance of the number of heads. c) The standard deviation of the number of heads, and d) the ratio of the standard deviation to the mean, as a decimal or a percent. (You could call this quantity the "relative error" or some such term.)
   
   

6. Solve using matrix methods of your choice the two systems of equations.
   
   
   
7. A fair die is rolled over and over again until either a six has been rolled or a three has been rolled twice. (Not necessarily consecutively.) Set this up as a Markov chain using the following 4 states:
   
   
   
   1. Draw the transition diagram for this and write down the transition matrix.
   2. Find the fundamental matrix (Remember, )
   3. (Bonus-10 quiz points) What is the probability you roll two threes before you roll a 6?
   
   
   

   
   

8. A Markov chain has the following transition matrix:
   
   
   1. Find the values of a and and write down the transition matrix again using these values.
   2. Find the stable probabilities for this Markov chain (i.e. fixed probability vector.)
      

   3. Given that , find the maximum possible value of using the simplex method.
      

   4. The following facts are given: Find the maximum value of . using the big-M version of the simplex method. (ANSWER: the maximum value of is 10 at the point (5,10); what is important here is the method.)