Homework 6

[Note: this webpage last modified Friday, 04-Feb-2011 19:44:51 EST]

This homework assignment will be handed out on Wednesday September 29 and is due by the beginning of class on the following Wednesday, so by 3pm Wednesday October 3. REMEMBER that your solutions should be explained in complete sentences such that your classmates will understand the solution (and can verify your proofs are correct) even if they have not solved the problems themselves.

  1. HW Polices.

    1. Type your HW (in word, latex, text file, etc.) and send to me by email.

    2. At the beginning of your HW, list who you collaborated with. List "none" if none.

    3. Do NOT share electronically. You must type your own solutions. You can discuss the problems with each other, but you may only discuss them. You may not write out solutions together.

    4. You MAY NOT search the internet, textbooks, etc. for solutions to the problems. The following are the ONLY sources of information that you may use in solving the problems: the textbook for this course (draft available from here ), Computational Complexity: A Modern Approach, and wikipedia articles on basic math/probability/etc. You may discuss the problems with each other and with myself, but must obey the previous item in doing so.

      If you do find the solution in one of these three sources, you still MUST cite the source in your document.

      You may use NOTHING ELSE that is online or other textbooks. You have already received warnings about this! ANY violations of this policy for the rest of the semester will result in you FAILING the course and being brought before the appropriate committee on campus for handling academic misconduct.

  2. (5 Points) Suppose we take a room of n people, where we assume their birthdays are taken independently at random from all possible 365 birthdays (ignoring leap years). Compute the expected value that at least 3 people all share the same birthday. You should prove/derive the value/formula using the basic rules of probability (e.g., linearity of expectation, dealing with a product of independent variables).

  3. This problem considers the use of DNA evidence in trials. Suppose you are on trial, and your DNA matches DNA taken from the crime scene that was left by the perpetrator. The DNA test has the property that there is a false positive (a match between people with different DNA) with probability 10-6. The test has the additional property that false negatives (a non-match between two samples of the same DNA) never occur.

    The prosecutor claims that we can therefore conclude that the probability you are innocent is 10-6. Assuming there is no other evidence in the case, is this true? That is what we aim to answer in this problem.

    1. (5 Points) Let D be the event that your DNA matches the DNA from the crime scene, and let I be the event that you are innocent of the crime. Suppose the city you live in has a population of 108, and we know with certainty that the perpetrator is one of these people. Assuming no evidence at all (not even the DNA evidence above), what is a reasonable value for Pr[I]? What is a correct value for Pr[D]?

      Hint: For Pr[D], use the law of total probability, explained here: http://en.wikipedia.org/wiki/Law_of_total_probability.

    2. (5 Points) Use Bayes' rule and your results from Part (a) to compute a value for Pr[I | D].

      Is the prosecutor correct in his/her claim?

  4. (15 Points) Show that if NP is contained in BPP, then NP is contained in RP. Equivalently, show that if SAT is in BPP, then SAT is in RP. So, given a two-sided error algorithm for SAT, you want to create a one-sided error algorithm.

    Hint: Given a formula phi, you want to try to find a satisfying assignment for the formula. If it is satisfiable, then you need to show you can find one with probability greater than 1/2. If it is not satisfiable, then you should always reject. The way I envision solving the problem requires using error reduction for the BPP algorithm in addition to using a union bound (for a description of that, see here: http://en.wikipedia.org/wiki/Boole%27s_inequality).