\fA) Suppose that you have 9 cards. 4 are green and 5 are yellow. The4green cards are numbered1, 2, 3, and 4. The5yellow cards are numbered 1, 2, 3, 4, and 5. The cards are well shuffled. Suppose that you randomly draw two cards, one at a time, and with replacement. G1=first card is green G2=second card is green 1) Draw a tree diagram of the situation. (Enter your answers as fractions.) (see previous attachment also) 2) Enter the probability as a fraction. P (G1 AND G2) = ? 3) Enter the probability as a fraction. P (at least one green) =? 4) Enter the probability as a fraction. P (G2 | G1) = 5) Are G2 and G1 independent events? Explain why or why not. a) G1 and G2 are not independent because after choosing the first green card, the second green card has less chance of being picked. b) G1 and G2 are independent events because choosing a green card and replacing it does not affect the chances of choosing a second green card. c) G1 and G2 are not independent because they are the same color card. d) G1 and G2 are independent events because the probability of choosing a green card each time is the same. B) The following table identifies a group of children by one of four hair colors, and by type of hair. 1) Complete the table. Brown Wavy 21 Straight 78 Blon d 15 Totals 21 Blac k Red Totals 16 Hair Type 3 46 10 216 2) What is the probability that a randomly selected child will have wavy hair? (Enter your answer as a fraction.) 3) What is the probability that a randomly selected child will have either brown or blond hair? (Enter your answer as a fraction.) 4) What is the probability that a randomly selected child will have wavy brown hair? (Enter your answer as a fraction.) 5) What is the probability that a randomly selected child will have red hair, given that he has straight hair? (Enter your answer as a fraction.) 6) If B is the event of a child having brown hair, find the probability of the complement of B. (Enter your answer as a fraction.) 7) If B is the event of a child having brown hair, what does the complement of B represent? a) The complement of B would be the event of a child having wavy or straight hair. b) The complement of B would be the event of a child having blond or black hair. c) The complement of B would be the event of a child not having brown hair. d) The complement of B would be the event of a child having blond hair. C) 1) Complete the PDF. x P(X = x) 0 0.1 1 0.2 x P(X = x) 2 3 0.4 2) Find the probability that X= 2. D) A school newspaper reporter decides to randomly survey 18 students to see if they will attend Tet festivities this year. Based on past years, she knows that 17% of students attend Tet festivities. We are interested in the number of students who will attend the festivities. 1) In words, define the Random Variable X. a) the number of students that will attend Tet b) the types of festivities offered at the school c) the number of students in the school d) the school newspaper 2) List the values that X may take on: a) X = 1, 2, 3, . . ., 18 b) X = 1, 2, 3, . . ., 17 c) X = 0, 1, 2, . . ., 17 d) X = 0, 1, 2, . . ., 18 3) Give the distribution of X X~__ (__, __) 4) How many of the 18 students do we expect to attend the festivities? (Round your answer to the nearest whole number of students) 5) Find the probability that at most 4 students will attend. (Round your answer to four decimal places.) 6) Find the probability that more than 3 students will attend. (Round your answer to four decimal places.) E) For each probability and percentile problem, draw the picture. The speed of cars passing through the intersection of Blossom Hill Road and the Almaden Expressway varies from 11 to 35 mph and is uniformly distributed. None of the cars travel over 35 mph through the intersection. 1) In words, define the Random Variable X. a) the number of cars driving below 35 mph through the intersection b) the time, in seconds, it takes a car to pass through the intersection c) the number of cars passing through the intersection d) the speed, in mph, of an individual car passing through the intersection 2) Give the distribution of X. X~ (_, _) 3) Graph the probability distribution. 4) Enter an exact number as an integer, fraction, or decimal. f(X) =___, where ____ X ____ 5) Enter an exact number as an integer, fraction, or decimal. = 6) = (rounded to 2 decimal places) 7) What is the probability that the speed of a car is at most 26 mph? (Enter your answer as a fraction.) 8) What is the probability that the speed of a car is between 17 and 23 mph? (Enter your answer as a fraction.) 9) State "P(22 < X < 51) = ___" in a probability question. What is the probability? 10) Find the 90th percentile. This means that 90% of the time, the speed is less than __mph while passing through the intersection. 11) Find the 75th percentile. In a complete sentence, state what this means. This means that___% of the time, the speed is less than ___mph while passing through the intersection. 12) Find the probability that the speed is more than 24 mph given (or knowing that) it is at least 14 mph. (Enter your answer as a fraction.) F) For each probability and percentile problem, draw the picture. Suppose that the useful life of a particular car battery, measured in months, decays with parameter 0.02. We are interested in the life of the battery. 1) In words, define the Random Variable X. a) the number of batteries that last a lifetime b) the number of car batteries that an individual needs to replace c) the life, in years, of a car battery d) the life, in months, of a car battery 2) Is X continuous or discrete? 3) Give the distribution of X. X~___ (___) 4) On average, how long would you expect 1 car battery to last in months? 5) On average, how long would you expect 5 car batteries to last, if they are used one after another in months? 6) Find the probability that a car battery lasts more than 37 months. (Round your answer to four decimal places.) 7) 60% of the batteries last at least how long in months? (Round your answer to two decimal places.)