Show your workings.
7E-1 A bridge hand in which there is no card higher than a nine is called a Yarborough. Specify an appropriate sample space and determine the prob- ability of Yarborough when you are randomly dealt 13 cards out of a well- shuffled deck of 52 cards. 7E-2 Five dots are placed at random on a 8 x 8 grid in such a way that no cell contains more than one dot. Specify an appropriate sample space and determine the probability that no row or column contains more than one dot. 7E-3 Five people are sitting at a table in a restaurant. Two of them order coffee and the other three order tea. The waiter forgot who ordered what and puts the drinks in a random order for the five persons. Specify an appropriate sample space and determine the probability that each person gets the correct drink 7E-4 A parking lot has 10 parking spaces arranged in a row. There are 7 cars parked. Assume that each car owner has picked at a random a parking place among the spaces available. Specify an appropriate sample space and determine the probability that the three empty places are adjacent to each other. 7E-5 Somebody is looking for a top-floor apartment. She hears about two vacant apartments in a building with 7 floors en 8 apartments per floor. What is the probability that there is a vacant apartment on the top floor? 7E-6 You choose at random two cards from a standard deck of 52 cards. What is the probability of getting a ten and hearts? 7E-7 A box contains 7 apples and 5 oranges. The pieces of fruit are taken out of the box, one at a time and in a random order. What is the probability that the bowl will be empty after the last apple is taken from the box? 7E-8 A group of five people simultaneously enter an elevator at the ground floor. There are 10 upper floors. The persons choose their exit floors inde- pendently of each other. Specify an appropriate sample space and determine the probability that they are all going to different floors when each person randomly chooses one of the 10 floors as the exit floor. How does the answer change when each person chooses with probability + the 10th floor as the exit floor and the other floors remain equally likely as the exit floor with a probability of 1 each. 7E-9 Three friends and seven other people are randomly seated in a row. Specify an appropriate sample space to answer the following two questions. (a) What is the probability that the three friends will sit next to each other?9E-12 You have a thoroughly shuffled deck of 52 cards. Each time you choose one card from the deck. The drawn card is put back in the deck and all 52 cards are again thoroughly shuffled. You continue this procedure until you have seen all four different aces. What are the expected value and the standard deviation of the number of times you have to draw a card until you have seen all four different aces? 9E-13 A group of m people simultaneously enter an elevator at the ground floor. Each person randomly chooses one of the y floors 1, 2, ..., " as the exit floor, where the choices of the persons are independent of each other. The elevator only stops on a floor if at least one person wants to exit on that floor. No other people enter the elevator at any of the floors 1, 2, ..., r. What are the expected value and the standard deviation of the number of stops the elevator will make? 9E-14 (a) An integer is repeatedly drawn at random from 1, 2, . .., 10. What are the expected value and the standard deviation of the number of integers from 1,2,..., 10 that do not show up in 20 drawings? (b) In each drawing of the Lotto 6/45 six different integers are randomly chosen from 1, 2, ..., 45. What are the expected value and the standard deviation of the number of integers from 1, 2, . .., 45 that do not show up in 15 drawings? 9E-15 Take a random permutation of the integers 1, 2, ..., n. Let us say that the integers i and j with i # j are switched if the integer i occupies the jth position in the random permutation and the integer / the ith position. What is the expected value of the total number of switches? 9E-16 Twelve married couples participate in a tournament. The group of 24 people is randomly split into eight teams of three people each, where all possible splits are equally likely. What is the expected value of the number of teams with a married couple? 9E-17 Suppose you know that the hands of you and your bridge partner contain eight of the 13 spades in the deck. What is the probability of 3-2 split of the remaining five spades in the bridge hands of your opponents? 9E-18 You choose at random an integer from 1, 2, ... ,6. Next you roll a fair die until you get an outcome that is larger than or equal to the randomly chosen integer. What is the probability mass function of the number of times you will roll the die? What are the expected value and the standard deviation of the number of times you will roll the die? 9E-19 You have two coins. One coin is fair and the other is biased with probability p of heads. The first toss is done with the fair coin. At the12E-1 Let Z, and Z, be independent random variables each having the standard normal distribution. Define the random variables X and Y by X = Z1 +3%2 and Y = 21 + Zz. Argue that the joint distribution of (X, Y) is a bivariate normal distribution. What are the parameters of this distribu- tion? 12E-2 Let the random vector (X, Y) have the standard bivariate normal distribution with correlation coefficient p = -0.5. What are the values of a for which the random variables V = aX + Y and W = X + aY are independent? 12E-3 Let the random vector (X, Y) have the standard bivariate normal dis- tribution with correlation coefficient p with -1 X [ X > 0) and P(Y/X
