Problem 1 Show that for any collection of...

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MATH 230/STAT 230: Homework 2 Due on 09/10, in class Read Section 1.1-Section 1.5 Problem 1. Show that for any collection of events { A i } n +1 i =1 , P n +1 [ i =1 A i = P n [ i =1 A i + P ( A n +1 ) - P n [ i =1 ( A i A n +1 ) . (1) Problem 2. (i) For two events A and B show that P ( A B ) P ( A ) + P ( B ) - 1 . (ii) (Boole’s inequality) For a collection of events { A i } n i =1 show that P n [ i =1 A i X i P ( A i ) . Hint: For (ii) you may use (1), and induction. Problem 3. A card is selected at random from a deck of 52 playing cards. If E is the event that the card is a King and F is the event that it is a heart. Show that E and F are independent events. Problem 4. An urn contains 4 white balls and 6 black balls. A ball is chosen at random, and its color is noted. The ball is then returned, along with 3 more balls of the same color. Then another ball is drawn at random from the urn. (i) Find the chance that the second ball drawn is white. (ii) Given the second ball drawn is white, what is the probability that the first ball drawn is black ? (iii) Suppose the original contents of the urn are w

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