Homework Assignment 1 Due in class, Thursday Jan 29 SDS 321 Intro to Probability and Statistics 1. Draw Venn diagrams compatible with the following statements: (a) A ∩ B = ∅ Ω A B (b) A ⊂ B, C ⊂ B, A ∩ C = ∅. Ω B A C (c) A ∪ B ∪ C = Ω, A ∩ B = ∅, A ∩ C = ∅, B ∩ C = ∅ C A B This should be pretty straightforward – drop me an email if there are any you don’t get 2. For two sets A and B, show that (A ∪ B)c = Ac ∩ B c Let x ∈ (A ∪ B)c . Then, x 6∈ A ∪ B, and since A ∪ B = {x|x ∈ A or x ∈ B}, x 6∈ A and x 6∈ B. Therefore, x ∈ Ac and x ∈ B c , therefore x ∈ Ac ∩ B c (since, Ac ∩ B c := {x|x ∈ B c and x ∈ Ac ). 1 3. I want to build a probability model for the delivery of a package, which is guaranteed to arrive some time next week. I want to construct a probability law describing the probability of it arriving on any given day. We can denote the probability that it arrives on Monday as p1 , the probability of it arriving on Tuesday as p2 , and so on. Assume that the arrival probability doubles every day, i.e. p2 = 2p1 , p3 = 2p2 , ... and p7 = 2p6 . Can you write down a probability Plaw for when the package arrives, which follows all the axioms? Hint: You can use 7i=1 2i−1 = 27 − 1. p1 = 20 p1 , p2 = 21 p1 , . . . , p7 = 26 p1 . Everything must normalize to 1, so pk = 2k−1 /(26 − 1) (or write down the individual probabilities) 4. For any two events A and B, show that P(A ∪ B) = P(A) + P(B) − P(A ∩ B) P (A ∪ B) = P (A ∩ B c ) + P (A ∩ B) + P (B ∩ Ac ). P (A) = P (A ∩ B) + P (A ∩ B c ) and P (B) = P (B ∩A)+P (B ∩Ac ), so P (A)+P (B) = 2P (A∩B)+P (A∩B c )+P (B ∩Ac ) = P (A ∪ B) + P (A ∩ B) 5. I toss a (fair) coin and roll a (fair, 6-sided) die. I am interested in the value from both the coin and the die. (a) What is an appropriate sample space? something like {H1, H2, . . . , H6, T 1, T 2, . . . , T 6} (b) What is the probability that the coin yields heads and the die yields 4? 1/12 either by uniform probability (12 elements in sample space) or by independence. (c) What is the probability that the die yields a number less than 3? 2/3 (d) What is the probability that the coin yields heads and the die yields an even number, OR the coin yields heads and the die yields an odd number? 1/2 6. I have two coins. I know that one is fair (heads and tails are equally likely), and the other is biased (P(H) = 0.7), but I don’t know which is which. I pick a coin at random, with each coin being equally likely. I toss the coin, and get tails. What is the probability that it is the biased coin? Using Bayes’ law: P (T |B)P (T ) P (T |B)P (T ) + P (T |B c )P (B c ) 0.3 3 0.3 × 0.5 = = = 0.3 × 0.5 + 0.5 × 0.5 0.3 + 0.5 8 P (B|T ) = 7. According to exit polls following the 2012 election, 53% of voters were women. 55% of female voters voted for Barack Obama, and 45% of male voters voted for Obama. Draw a tree representing the conditional probabilities, and use this to calculate the probability that a randomly selected voter is a man who did not vote for Obama. Draw a tree where the first branch is woman/man, second branch is Obama|woman, Not Obama|woman, etc. Get joint probabilities by following the branches - so follow “man” - P (man = 0.47 then Obama|man - P (Obama|man) = 0.45 so that P (man∩Obama = 0.47 ∗ 0.45 = 0.2115. 2