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?(b) An insect breeding experiment was conducted in two sections, in each of which 100 insects of a particular species were raised. In the first section a proportion p was expected to have a certain colour variation and in the second section the proportion with this colour variation was expected to be p', but the value of p was not known. In the event there were 22 insects in the first section, and 7 in the second section, which possessed the colour variation. Find the value of p which maximises the probability of this result.1B.11 Length of a chord of a circle In Figure 1.6, OP is a radius of a circle with centre O and radius r. A point Q is chosen on OP in such a way that all points on OP have the same chance of being chosen. The chord passing through Q at right angles to OP meets the circle at points A and B. Find the probability density function of the length AB , and show that the probability that AB is greater than r is 0.866. 0 X B Figure 1.6