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?probability : for i = 0, 1,....". You choose at random one of the coins. Use Bayes' rule in odds form to find the probability that you have chosen coin's given that each of the first n tosses has resulted in heads. 8E-32 Your friend has generated two random numbers from 1. ..., 10, inde- pendently of each other. Use Bayes' rule in odds form to answer the following two questions. (a) What is the probability that both numbers are even given the information that there is an even number among the two numbers? (b) What is the probability that both numbers are even given the information that the number 2 is among the two numbers? 8E-33 Your friend has fabricated a loaded die. In doing so, he has first simulated a number at random from 0.1, 0.2, 0.3, and 0.4. He tells you that the die is loaded in such a way that any roll of the die results in the outcome 6 with a probability which is equal to the simulated number. Next the die is rolled 300 times and you are informed that the outcome 6 has appeared 75 times. What is the posterior distribution of the probability that a single roll of the die gives a 6? 8E-34 Your friend is a basketball player. To find out how good he is in free throws, you ask him to shoot 10 throws. You assume the three possible values 0.25, 0.50 and 0.75 for the success probability of the free shots of your friend. Before the 10 throws are shot, you believe that these three values have the respective probabilities 0.2, 0.6 and 0.2. What is the posterior distribution of the success probability given that your friend scores 7 times out of the 10 throws?10E-27 The lifetimes of two components in an electronic system are inde- pendent random variables X, and X2, where X, has a normal distribution with an expected value of , time units and a standard deviation of of time units. What is the probability that the lifetimes of the two components expire within a time units from each other? 10E-28 A space mission will take 150 days. A number of copies of a daily-use appliance must be taken along. The amount of time the appliance can be used is exponentially distributed with an expected value of two days. What is the probability mass function of the number of copies of the appliance to be used when an infinite supply would be available? Use the normal approximation to find how many copies of the appliance should be stocked so that the probability of a shortage during the mission is no more than 10-3. Compare the approximate result to the exact result. 10E-29 A new casino has just been opened. The casino owner makes the following promotional offer to induce gamblers to play at his casino. People who bet $10 on red get half their money back if they lose the bet, while they get the usual payout of $20 if they win the bet. This offer applies only to the first 2,500 bets. In the casino European roulette is played so that a bet on red is lost with probability - and is won with probability . Use the normal distribution to approximate the probability that the casino owner will lose more than 6,500 dollars on the promotional offer. 10E-30 Consider again Problem 10E-29. Assume now that the casino owner makes an offer only to the most famous gambler in town. The casino owner lends $1,000 to the gambler and proposes him to make 100 bets on red with this starting capital. The gambler is allowed to stake any amount of his bankroll at any bet. The gambler gets one fourth of the staked money back if he loses the bet, while he gets double the staked money back if he wins the bet. As reward, the gambler can keep to himself any amount in excess of $1,000 when stopping after 100 bets. Suppose that the gambler decides to stake each time a fixed percentage of 5% of his current bankroll. Use the normal distribution to approximate the probability that the gambler takes home more than d dollars for d = 0, 500, 1,000 and 2,500. Hint: consider the logarithm of the size of the bankroll of the gambler. 10E-31 A battery comes from supplier 1 with probability p, and from supplier 2 with probability pz, where pi + p2 = 1. A battery from supplier i has an exponentially distributed lifetime with expected value 1/p; for i = 1, 2. The This problem is related to Kelly betting discussed in the Sections 2.7 and 5.9 of the book. 19 battery has already lasted s time units. What is the probability that the battery will last for another { time units? 10E-32 The failure rate function of the lifetime of a vacuum tube is r() = (me mite "-(tuz)e (it))/(emItem-e (it)) for a > 0, where 0
