## hw1

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## hw1

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- Pages:
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- School:
- University of California, Los Angeles
- Course:
- Stats 100c - Linear Models

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170A Killip Homework 1 Due Fri Sept 30 1 A fair die is rolled and a fair coin is flipped a Write down a sample space for this experiment b Write down the event as a set that the coin lands heads c Name the events which have the smallest and largest probabilities d Write down the following event and compute its probability the coin is heads and the die roll is odd but not three or the die roll is three e Careless writing makes the following description ambiguous The die is 3 or even and the coin is heads or the coin is tails or the die is 4 Write all possible interpretations i e distinct answers stemming from adding the needed brackets and their probabilities 2 If the sample space has n elements what is the total number of possible events 3 A fair coin is tossed and a subset of 0 1 2 is chosen at random All outcomes are equally likely a Write down a suitable sample space b What is the probability that the chosen subset contains the element 0 4 A number is to be chosen at random from the range 0 x 1 a Choose an appropriate sample space b We write x as a decimal without recurring 9 Write down the event that the first or second digit after the point is a three 5 A number is to be chosen at random from the range 0 x 1 It is given that P x a x b b a for every pair 0 a b 1 Use Problem 13 and the axioms to show a P x 0 x 12 21 b P x x 31 0 6 Problem 6 from Chapter 1 7 Problem 8 from Chapter 1 8 a Starting from the axioms show that for any events A1 A2 An we have P A1 A2 An P A1 P A2 P An We will refer to this inequality as the union bound b Use this result to resolve Problem 11 from Chapter 1 Continued 1 2 9 Study problem 12 from Chapter 1 10 Suppose we choose an integer 1 n 10 at random where each answer is equally likely Compute the probability that n is a prime number using inclusion exclusion and the fact that n is not prime n 1 or n is a multiple of 2 other than 2 itself or n is a multiple of 3 other than 3 or n is a multiple of 5 other than 5 or

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