UNIVERSITY OF WESTERN ONTARIO Computer Science 2214a, Fall 2016 - 2017 Discrete Structures for Computing ASSIGNMENT 5 Given: Wednesday Nov.30, Due: Wednesday Dec. 7, 6:00pm 1. Suppose you randomly pick a bit string from the set of all bit strings of length 12. a) What is the probability that the bit string begins and ends with 1? b) What is the probability that the bit string has exactly two 1s, given that the string begins with a 1? c) What is the probability that the bit string has more 1s than 0s? d) What is the probability that the bit string has the sum of its digits equal to seven? e) What is the probability that the bit string begins with 110? Provide detailed justifications for your answers. 2. What is the probability of these events when we randomly select a permutation of the 26 lowercase letters of the English alphabet? (a) The first 12 letters in the permutation are in alphabetical order. (b) a is the first letter of the permutation and z is the last letter. (c) a and z are next to each other in the permutation. (d) a and z are separated by at least 23 letters in the permutation. (e) z precedes both a and b in the permutation. Provide detailed justifications for your answers. 3. Suppose that 3% of the patients tested in a clinic have a peanut allergy. Furthermore, suppose that when a blood test for peanut allergy is given, 98% of the patients who have peanut allergy test positive, and that 5% of the patients who do not have peanut allergy test positive. Use Bayes' Theorem to find what is the probability that: (a) a patient testing positive for peanut allergy with this test is actually allergic to peanuts. 1 (b) a patient testing positive for peanut allergy with this test is not allergic to peanuts. (c) a patient testing negative for peanut allergy with this test is not allergic to peanuts. (d) a patient testing negative for peanut allergy with this test is allergic to peanuts. Provide detailed justifications for your answers. 4. Suppose that a Bayesian spam filter is trained on a set of 400 spam messages and 2000 messages that are not spam. The word \"incredible\" appears in 40 spam messages and in 250 messages that are not spam, and the word \"opportunity\" allears in 60 spam messages and in 300 messages that are not spam. Would an incoming message be rejected as spam if it contains both the word \"incredible\" and the word \"opportunity\