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ProblemsDirections: For all problems where appropriate define your random variable, specify the distribution of therandom variable, and convert the English prose into a probability statement.1. If two fair dice are rolled 10 times, what is the probability of at least one 6 (on either die) in exactlyfive of these 10 rolls? 0.10840242. On a random day, the number of vacant rooms of a big hotel in New York City is 35, on average. Whatis the probability that next Saturday this hotel has at least 30 vacant rooms? 0.82295453. Suppose that X is a Poisson random variable with P (X = 1) = P (X = 3). Find the P (X = 5).0.063444944. An absentminded professor does not remember which of his 12 keys will open his office door. If hetries them at random and with replacement:a. On average, how many keys should he try before his door opens? 12b. What is the probability that he opens his office door after only three tries? 0.070023155. Suppose that 20% of a group of people have hazel eyes. What is the probability that the eighthpassenger boarding a plane is the third one having hazel eyes? Assume that passengers boarding theplane form a randomly chosen group. 0.055050246. From a panel of prospective jurors, 12 are selected at random. If there are 200 men and 160 womenon the panel, what is the probability that more than half of the jury selected are women? 0.24440037. It takes a professor a random time between 20 and 27 minutes to walk from his home to school everyday. If he has class at 9:00 A.M. and he leaves home at 8:37 A.M., find the probability that he reacheshis class on time.378. Suppose that a Scottish soldier’s chest size is normally distributed with mean 39.8 and standarddeviation 2.05 inches, respectively. What is the probability that of 20 randomly selected Scottishsoldiers, five have a chest of at least 40 inches? 0.030321009. Suppose that lifetimes of light bulbs produced by a certain company are normal random variables withmean 1000 hours and standard deviation 100 hours. Is this company correct when it claims that 95%of its light bulbs last at least 900 hours?10. The lifetime of a TV tube (in years) is an exponential random variable with mean 10. If Jim boughthis TV set 10 years ago, what is the probability that its tube will last another 10 years? 0.367879411. In a hospital, babies are born at a Poisson rate of 12 per day. What is the probability that it takes atleast seven hours before the next three babies are born? 0.320847212. The grades of students in a calculus-based probability course are normal with a mean of 72 and standarddeviation 7. If 90, 80, 70, and 60 are the respective lowest, A, B, C, and D, what percent of students inthis course get A’s, B’s, C’s, D’s, and F’s? 0.005063995, 0.12148050, 0.4859026, 0.3443103, 0.0432381313. Alan and Susie play a series of backgammon games until one of them wins five games. Suppose thatthe games are independent and the probability that Susie wins is 0.58.1

a. Find the probability that the series ends in seven games. 0.2396188b. If the series ends in seven games, what is the probability that Susie wins? 0.724784514. A certain basketball player makes a foul shot with probability 0.45. Determine for what value of k theprobability of k baskets in 10 shots is maximized, and find this maximized probability.15. A restaurant serves eight fish entrées, 12 beef, and 12 poultry. If customers select from these entréesrandomly, what is the expected number of fish entrées ordered by the next four customers? 116. Suppose that a certain bank returns bad checks at a Poisson rate of three per day. What is theprobability that this bank returns at most four bad checks during the next two days? 0.285056517. When a certain car breaks down, the time that it takes to fix it (in hours) is a random variable withthe density function f(x) = ce −3x if 0 ≤ x < inf and 0 otherwise.a. Calculate the value of c. 3b. Find the probability that when this car breaks down, it takes at most 30 minutes to fix it. 0.776869818. The breaking strength of a certain type of yarn produced by a certain vendor is normal with mean 95and standard deviation 11. What is the probability that, in a random sample of size 10 from the stockof this vendor, the breaking strength of at least two are over 100? 0.885468619. If X is the number of 6’s which appear when 72 dice are thrown, what is the expected value of X 2 ?15420. The moment generating function of a random variable X is e 4(et −1) . Show that P (µ − 2σ < X