1. There are three sections of English 101. In Section I, there are 25 students, of whom 5 are mathematics majors. In Section II, there are 20 students, of whom 6 are mathematics majors. In Section III, there are 35 students, of whom 5 are mathematics majors. A student in English 101 is chosen at random. Find the probability that the student is from Section I, given that they are a mathematics major?
2. Twenty percent of the contestants in a scholarship competition come from Pylesville High School, 40% come from Millerville High School, and the remainder come from Lakeside High School. Two percent of the Pylesville students are among the scholarship winners; 3% of the Millerville contestants and 5% of the Lakeside contestants win.
(a) If a winner is chosen at random, what is the probability that they are from Lakeside?
(b) What percentage of the winners are from Pylesville?
Use Bayes' theorem to calculate the probabilities?