UCLA STATS 100C - hw3 (2 pages)

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170A Killip Homework 3 Due Fri Oct 14 1 Problem 31 from Chapter 1 2 Study Problem 42 from Chapter 1 3 Problem 40 from Chapter 1 4 Suppose my knowledge ignorance of the number of branches of a certain store is given by the following probability law P k branches 1 p pk where 0 p 1 and k 0 1 2 3 If I subsequently discover that they have at least 7 branches e g I walk into store and it says branch 7 what new probability law describes my revised knowledge 5 Here are the probabilities for the outcomes in the last problem on HW2 M A E E F A T 1 16 7 32 T 1 8 1 8 N 3 32 1 8 N 3 16 1 16 Show that F and T are not independent but are independent conditioned on A 6 Which is more probable to obtain n heads from tossing a fair coin independently 2n times or to obtain n 1 heads by throwing the coin 2n 2 times Compute the exact ratio of these probabilities 7 Starting at the origin on the line we take a step of one unit to the left or to the right with probability 1 2 We do this repeatedly with independent steps If we take 2n steps what is the probability that we find ourselves back at the origin 8 Problem 53 from Chapter 1 9 Problem 58 from Chapter 1 10 Seven blue and four red balls are to be arranged in order How many ways can this be done if a The blue balls are distinguishable e g numbered as are the red balls b Blue balls are distinguishable but the red balls are identical c The balls of each color are indistinguishable 11 How many ways can we order the twenty six letters of the alphabet together with seven indistinguishable symbols Continued 1 12 How many ways can we distribute n balls among k bags if a the balls and bags are distinguishable e g numbered b the bags are distinguishable the balls are not c balls and bags are distinguishable but the bags can contain at most one ball necessarily k n d the bags are distinguishable the balls are not and the bags can contain at most one ball Hint The previous question has the same answer as b with n 26 and k 8