1) A bin contains a total of 15 balls: 4 yellow, 2 red, and 9 blue. Two balls are selected without replacement. (8 marks) a ) What is the probability of selecting at least one yellow ball? What is the probability of selecting exactly one yellow ball? b) What is the probability of selecting two blue balls? How does the probability change if the first ball is put back in the bin before selecting a second?2 ) A pair of standard 6 sided dice is rolled. The two dice are independent and labelled with the numbers 1 to 6. (10 marks) a) What is the most likely outcome and what is the probability of that outcome? b ) After 5 rolls (of two dice), what is the probability of rolling an 11 at least once? C ) Sketch the probability distribution and describe its shape. d) Compute the mean and standard deviation of a single, two dice roll.3 ) You are dealt two cards in a game of Texas Hold'em and have no knowledge of any other cards. (8 marks) a ) What is the probability of being dealt two spades? What is the probability that your cards are suited? b ) What is the probability of being dealt a pair? c) What is the probability of being dealt no face cards and no aces.4) Battery Brigade has tested 16 Die Hard car batteries, each five years old, to see whether each battery will start a car after sitting out in -30 C weather for 24 hours. On the basis of past experience, Battery Brigade estimates the probability to be 0.05 that a battery will not start the car. Use the binomial probability distribution to answer the following questions, rounding final answers to 4 decimal places (8 marks) a) What is the probability that 3 of the batteries will not start the car? b ) What is the probability that all 16 of the batteries will start the car? C ) What is the probability that 2 or more of the batteries will not start the car? d) Compute the mean and standard deviation of this binomial probability distribution.5) Use the following contingency table which summarizes the results of a study to determine if there is a relationship between place of residence and ownership of a foreign car. (8 marks) Ownership Place of Residence Foreign (F) | Domestic (D) | Total Large City (L) 90 110 200 Suburbs (S) 60 90 150 Rural (R) 50 100 150 Total 200 300 500 If one of the 500 car owners is randomly selected ... a ) What is the probability of selecting a foreign car owner? b ) What is the probability of selecting a foreign car owner or someone who lives in the suburbs? c) What is the probability of selecting a domestic car owner given that she or he is from a rural area? d) Determine whether or not S and F are independent events.6) A smoke detector system uses two devices, an ionization sensor and a photoelectric sensor. If smoke is present, the probability that it will be detected by the ionization sensor (1) is 0.90; by the photoelectric sensor (P), 0.80; and by both devices, 0.72. (8 marks) a ) Draw a Venn diagram to represent the problem b ) If smoke is present, determine the probability that the smoke will be detected by the system. C ) If smoke is present, determine the probability that the smoke is undetected. d) Are the two alarms operating independently. Justify your answer with a calculation