CMPSCI 190DM: A Mathematical Foundation for Informatics

Practice for Final Exam

David Mix Barrington

2 December 2013

Directions:

• Answer the problems on the exam pages.
• There are seven problems for 125 total points. Scale will be determined after the exam but a good guess is A = 105, C = 75.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

  Q1: 20 points
  Q2: 15 points
  Q3: 15 points
  Q4: 20 points
  Q5: 15 points
  Q6: 20 points
  Q7: 20 points
Total: 125 points

• Question 1 (20): Briefly identify the following terms or concepts (4 points each):
  • (a) a proof by contradiction

  • (b) the intersection of two sets A and B

  • (c) a bijection between two sets A and B

  • (d) a permutation from a set A

  • (e) the expected value of a random variable

• Question 2 (15): In this question p, q, and r are three boolean variables. We are going to pick the truth values of each variable by tossing a fair coin. The three tosses are assumed to be independent.
  • (a, 5) What is the probability that the compound statement "p → q" will be true, given the randomly chosen values of p, q, and r?

  • (b, 10) What is the probability that the compound statement "(p → q) ∧ (q → r)" will be true, given the randomly chosen values of p, q, and r?

• Question 3 (15): Let X be a set of n elements and let Y be a set of 3 elements. Prove by induction on n, starting with either n = 0 or n = 1 as you choose, that there are exactly 3ⁿ functions from X to Y.

• Question 4 (20): Suppose we choose a set of three cards randomly from a standard 52-card deck, with every three-card set being equally likely.
  • (a, 5) How many possible three-card subsets are there? (Remember that you need not evaluate an arithmetic expression.)

  • (b, 5) How many of these subsets have three cards of the same rank (for example, three fours or three kings)?

  • (c, 10) What is the probability that the three cards in the set all come from the same suit (for example, three hearts or three spades)?

• Question 5 (15): Suppose I throw a fair six-sided die n times, with each throw being independent of the others. Let p(n) by the probability that I ever get the same number twice in a row during these throws. Prove by induction on n, starting with n = 1, that p(n) = 1 - (5/6)^n-1. (Hint: You may find it easier to work with 1 - p(n).) Problem corrected on 5 December.

• Question 6 (20): Let A be the following matrix:

  
          0  1  1
          1  1  0
          0  2  0
        

  • (a, 5) Draw the directed multigraph whose matrix is A. Name the vertices x, y, and z, where the rows and columns correspond to the three vertices in that order.

  • (b, 5) Compute the matrix A².

  • (c, 10) In the directed graph of part (a), how many four-step paths are there from vertex y to itself? Explain your answer. (Hint: It is possible to count these paths by inspection of the graph, but that is not the easiest way to do it.)

• Question 7 (20): A prankster has modified a traffic light so that it behaves randomly instead of following its normal cycle. After a minute of being green, it becomes either yellow or red, with 50% probability of each color. After a minute of being yellow, it either becomes green or stays yellow with equal probability. After a minute of being red, it always becomes yellow.
  • (a, 5) Write down the transition matrix T for the Markov process that the light is following. (Order the rows and columns green-yellow-red.)

  • (b, 5) Compute the matrix T².

  • (c, 5) If the light is now red, what is the probability that it will also be red two minutes from now?

  • (d, 5) If the light is now red, what is the probability that it will also be red three minutes from now? Explain your answer.

Last modified 12 December 2013