This homework addresses the material covered in Lecture 2, on §1.3 in the text [1]. Recommended supplemental problems from text. These problems are not to be handed in with your assignment. You are encouraged to work with your classmates on the supplemental problems. • §1.3. Conditional probability: Problems 14, 15, 16, 17. Required problems for homework: 1. Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a recent study, suppose that pl = 16% of all ticks in a certain location carry Lyme disease, ph = 10% carry HGE, and pb = 10% of the ticks that carry at least one of these diseases in fact carry both of them. If a randomly selected tick is found to have carried HGE, what is the probability that the selected tick is also a carrier of Lyme disease? Hints: i) solve the general problem with pl = P(L),ph = P(H),pb, where L,H are the events of having Lyme, HGE, respectively; ii) note pb = P(L∩H|L∪H); iii) express pb in terms of P(L∩H) and P(L∪H); iv) use P(L∪H) = P(L)+P(H)−P(L∩H). 2. A class consisting of m graduate and n undergraduate students is randomly divided into m groups of (m + n)/m each (suppose m,n are such that (m + n)/m is an integer). What is the probability that each group includes a graduate student? 3. Generalize the analysis of Example 1.12 (The Monty Hall Problem) from 3 to n ≥ 3 doors. The prize is equally likely to be found behind any one of the n doors. After indicating a door, one of the remaining n−1 doors not containing the prize is opened. You may either i) stay (with the original door you picked) or ii) switch (to one of the remaining n−2 doors). Find the probability of winning under the stay and switch strategies as a function of n. 4. Consider a three by three grid with nine squares. Let m squares be selected uniformly at random, where m ∈{3,4,5,6}. Say a set of m squares is a bingo set if there exists a (length 3) row, column, or diagonal that is all in the set. Let p(m) be the probability of a bingo set. Write a computer program to randomly select an m set and evaluate whether or not it is a bingo set. Use this program to estimate p(m) by running k = 10,000 independent trials for each m ∈{3,4,5,6}. Produce ONE plot with four data sets: the running estimate of p(m) for each of the four values of m versus the number of trials k, for k = 1,2,...,10000.
