M230

Final Exam

Fall 1997

Bellamy

Problem 4 is worth twenty points and problem 8 is worth twenty bonus quiz points. The other problems are worth thirty points each.


  1. Compute each of the following matrix sums and products which can be computed; state which one are impossible:
    (a) (b)
    (c) (d)
    (e) (f)

  2. You know that x + 2y = 20 and x + y + z 18 where x, y, z 0. Find the maximum possible value of z using the big-M method (with M = 100 if you wish). Half credit for the correct answer by any other method.

  3. A student flying club had offered to fly six light airplanes from an airport in Maryland to one in New Jersey for a flying school. There are four single engine airplanes and two twin engin ones. There are ten students who are rated to fly single airplanes only and three others who are rated to fly either singles or twins. How many different ways can pilots be assigned to each airplane to transport them. (It does matter which pilot gets which airplane?) (Hint: Assign the pilots for the twins first.) If all these assignments are equally probable, what is the probability that none of the twin-engine rated pilots ends up flying a single?

  4. Given that x + y = 8 and x - y = 2, solve for x and y using each of the three methods we have studied (ten points for b); five each for a) and c)).

    1. Row reduction of the augmented matrix.
    2. Computing the inverse of the coefficient matrix and multiplying it by the column vector of constants.
    3. Ordinary non-matrix algebra calculations of your choice.


    Problems 5,6,7 and 8 all concern the following urns.


    Red urn: contains five red balls only.

    Green urn: contains three green balls only.

    Yellow urn: contains 4 yellow balls and one ball each of red, green, and blue.

    Blue urn: contains one red, two green, and four yellow balls.



  5. One ball is chosen randomly from each of the urns, independently, one after another. Let X denote the number of red balls drawn. Set up a table of possible values of X and their respective probabilities. Then compute E(X) as a fraction in lowest terms (It will be an improper fraction.)

  6. A ball is drawn randomly from an urn, its color noted, and it is replaced in the urn. Then a ball is drawn from the urn matching the previous ball. This process is repeated indefinatly. This is a Markov chain with four states, two of which, red and green, are absorbing.

    1. Draw the transition diagram for this Markov chain.
    2. Write down the transition matrix.
    3. If you first draw is from the yellow urn, what is the probability you eventually wind up in the red one?

    (This requires the recipe for absorbing Markov chains.)

  7. One of the urns is choosen randomly and a ball is drawn from it at random. Given that the ball was green, find the conditional probability that is came from the green urn. (Note: This is not a Markov chain problem!)

  8. (BONUS- 20 QUIZ POINTS) Use only the blue and yellow urns, and remove the red and green balls before we being the process in problem 6. This is a regular Markov chain. Write down the transition matrix and find the fixed probability vector, representing the long-term probabilities of drawing from the blue urn or the yellow urn.
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