Math 170A - Homework 1 - Due 4 October 2016 1. Let A = {1, 3, 4, 5}, B = {2, 4, 6, 7} and C = {1, 4}. Find (A B) C c . 2. Suppose that a 1 meter glass rod falls onto the floor and is broken into two pieces. Assume that the breaking point is uniform between 0 and 1 meter. What is the probability that the longer piece has length at least 0.6 meter? 3. Suppose that A and B are two events such that P(A) = 0.4, P(B) = 0.5 and P(AB) = 0.6. Find P(A B). 4. Let A, B, C be sets. Show that (i) A B c = A (A B)c and (ii) A C c (A B c ) (B C c ). 5. If A1 and A2 are two sets, we define the symmetric difference A1 A2 by A1 A2 = (A1 Ac2 ) (A2 Ac1 ). (This is simply the set of all points that either in A or B but not both.) Show that (i) P(A1 A2 ) = P(A1 ) + P(A2 ) 2P(A1 A2 ) and (ii) for every sets A, B, C, P(AC) P(AB) + P(BC). 6. Suppose that I arrive at a bus stop at some time between 8:00 am and 8:25 am with uniform probability. Buses arrive at 8:03, 8:13, 8:23, 8:33,. . .. What is the probability that I need to wait more than 2 minutes? 1