University of California, Berkeley, Statistics 134: Concepts of ProbabilityMichael Lugo, Fall 2011Exam 1September 28, 2011, 12:10 pm - 1:00 pmName:Student ID:This exam consists of eight pages: this cover page; six pages containing problems; anda table of the normal distribution. You may use a calculator, and notes on one side of astandard 8.5-by-11-inch (or A4) sheet of paper which you have written by hand, yourself.Questions 1 and 2 are multiple choice questions. You do not need to show your work onthese questions.For questions 3 through 7, you must show all work other than basic arithmetic. (You areallowed to use a calculator to do arithmetic, including binomial coefficients, factorials, andΦ, but you should indicate what you’re asking the calculator to do.) Give your answer toeach part as a fraction or decimal; your answers should not contain any binomial coefficients,factorials, or Φ.Write your name at the top of each page.The total number of possible points is 41. The number of points for each problem areindicated in brackets.DO NOT WRITE BELOW THIS LINE1. /1 5. /82. /1 6. /73. /9 7. /104. /5TOTAL /411Name:For the following two multiple-choice questions, circle the correct answer. No explanationis necessary.1. [1] Consider the following events:(i) in flipping 12 fair coins, exactly 6 heads are obtained.(ii) in flipping 108 fair coins, either 53, 54, or 55 heads are obtained.Which of the following statements is true?(a) The event (i) is considerably more likely than the event (ii).(b) These two events have approximately the same probability.(c) The event (ii) is considerably more likely than the event (i).2. [1] Say P (A|B) < P (B|A). Which of the following is true?(a) P (A) < P (B).(b) P (A) = P (B)(c) P (A) > P (B)(d) There is not enough information to tell.2Name:3. [total 9] I toss a fair coin repeatedly, until the third time it comes up heads, and Irecord the total number of coin tosses. For example, if I obtain the results T T HHT T H, Irecord the number 7. What is:(a) [2] the probability that this experiment requires exactly 7 tosses?(b) [2] the probability of observing the exact sequence T T HHT T H, given that the ex-periment required seven tosses?(c) [2] the probability that this experiment requires at least 8 tosses?(d) [3] I toss a coin repeatedly, until the 30th time it comes up heads, and I record thetotal number of coin tosses. What is the approximate probability that this requires at least70 tosses?3Name:4. [total 5] The events A, B, and C are independent. P (AB) = 0.4, P (AC) = 0.2, andP (ABC) = 0.1.(a) [2] What are the probabilities of A, B, and C?(b) [3] What is the probability that exactly one of A, B, and C occurs? (If you couldn’tsolve the previous question, give a formula for the answer to this question in terms of P(A),P(B), and P(C).)4Name:5. [total 8] In a distant land, an evil zookeeper named Theo has two barrels. Barrelnumber one contains three hippos and four monkeys. Barrel number two contains five hipposand three monkeys.In this problem, whenever something is chosen at random (a barrel, or an animal fromthat barrel), all the choices are equally likely to be chosen.(a) [2] Theo selects a barrel at random, and then removes a random animal from thebarrel. What is the probability that the animal is a hippo?(b) [3] Theo selects a barrel at random, and then removes a random animal from thebarrel. It is a hippo. What is the probability that the animal Theo removed came frombarrel number one?(c) [3] Theo selects a barrel at random, and then removes a random animal from thebarrel. It is a hippo. He does not put the hippo back in its barrel. Theo now removesanother animal from the same barrel. What is the probability, given that the first animal isa hippo, that the second animal Theo removes is a monkey?5Name:6. [total 7] I roll a fair die 500 times. What is the approximate probability that I obtain(a) [2] exactly 100 sixes?(b) [2] at least 95 sixes?(c) [3] I carry out this experiment 200 times. What is the approximate probability thatI roll exactly 100 sixes at least twice?6Name:7. [total 10] I have three dice. One is six-sided, one is eight-sided, and one is twelve-sided.The n-sided die has numbers 1 through n on its sides, and each side is equally likely to comeup. I roll the three dice.(a) [2] What is the probability that I roll the same number on all three dice?(b) [2] What is the probability that the product of the three numbers I roll is even?(Hint: the product of two odd numbers is odd; the product of an even number with anyother number is even.)(c) [3] What is the probability that the smallest number I roll is a 3?(d) [3] I pick one of the dice at random, with all three equally likely to be picked. I rollit and it comes up 4. What’s the probability the die I rolled was the six-sided