Question 5 (15 Points): Conditional probability (01,03) A student wrote a (machine learning) algorithm for distinguishing between pictures of cats and pictures of dogs. The error rates of her algorithm are given as: P(alg = dog|pic 2 cat) = 0.1 P(alg = cet|pic 2 dog) 2 0.2. [For example, P(ag = eet|pic 2 dog) denotes the probability that the algorithm outputs \"eat\" when seeing a picture of a dog] Now we test the algorithm for several rounds, each round inputting to it a random picture of either a cat or a dog. For each picture, the probability of it being a cat's picture is 0.3 and the probability of it being a dog's picture is 0.7. a} What is the probability that the algorithm correctly classies 3 consecutive pictures? (5 points) b) When the algorithm outputs \"ca \" , what is the probability that the picture actually shows a cat? (5 points} c) Suppose we run this test for 5 rounds, and we award the student 10 HKD for each picture that her algorithm correctly classies (and 0 HKD for each wrong classication). What is the variance of her income? (5 points} Question 6 (15 Points): A hard-working student (02,03) For 20 days, a student solves at least one exercise of discrete mathematics per day. We also know that the total number of exercises solved by the student is no more than 35. Prove or disprove the following statement: There must exist integers i and j, with i > j, such that the student has solved exactly 7 exercises between the end of day 5." and the end of day