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c Math 141, Fall 2014, Dr. Kamran Reihani - Texas A&M University Week in Review, Review for Exam 3 Disclaimer: This review is more heavily weighted on 8.1-8.4, although some problems from Chapter 7 are included. Please also take a look at the previous Week in Reviews for more practice problems on other sections. 1. Let E and F be two events, and suppose that P (E) = 0.35, P (F ) = 0.55 and P (E ∪ F ) = 0.70. (a) Compute P (E|F ). (b) Are the events E and F independent? 2. Five people are selected at random. (a) What is the probability that none of the people in this group were born on the same day of the week? (b) What is the probability that at least two people in this group were born on the same day of the week? 3. The US Senate consists of two senators from each of the 50 states. Five senators are to be selected at random to form a committee. What is the probability that no two members of the committee are from the same state? 1 c Math 141, Fall 2014, Dr. Kamran Reihani - Texas A&M University 4. For the following random variables, list the values that X can assume and state whether it is finite discrete, infinite discrete, or continuous. (a) Let X be the number of times it takes for you to hit the bull’s eye while playing darts. (b) Let X be the amount of water (in liters) that a person drinks each week. (c) Cards are drawn from a standard deck of cards with replacement until a king is drawn. Let X be the number of draws needed. (d) Bingo numbers are pulled out of a bag without replacement until B1 is called. Let X be the number of pulls needed. (There are 75 total bingo numbers.) (e) Three marbles are pulled in succession with replacement from a jar containing 3 red, 4 blue, and 5 green. Let X be the number of blue marbles pulled. (f) Let X be the distance (in miles) between College Station and a student’s hometown. 5. Two 4-sided die are rolled and the numbers are observed. Let X be the product of the numbers rolled. (a) Find the sample space for this experiment. 2 c Math 141, Fall 2014, Dr. Kamran Reihani - Texas A&M University (b) What values can X take on? (c) Find the probability distribution for X. (d) Draw a histogram of this distribution. (e) What is P (X ≥ 8)? 6. Data is given below showing the number of cars that are of a certain age (in years). The data was taken from a study of 2000 cars on a university campus. Age of Car 0 1 2 3 4 5 6 7 8 9 10 Number of Cars 140 350 450 650 200 120 50 10 5 15 10 (a) Find the mean, median, mode, variance, and standard deviation for the age of a car on this campus. (b) Let X be the age of a randomly selected car. Find the probability distribution for X. 3 c Math 141, Fall 2014, Dr. Kamran Reihani - Texas A&M University 7. A player pays $3 to play the following game: he tosses three fair coins and receives back “payoffs” of $1 if he tosses no heads, $2 for one head, $3 for two heads, and $4 for three heads. (a) Find the player’s expected net winnings for this game. (b) What cost would make this a fair game? 8. A man wishes to purchase a 5-year term-life insurance policy that will pay the beneficiary $20, 000 in the event that the man’s death occurs during the next 5 years. Using life insurance tables, he determines that the probability that he will live another 5 years is .96. What is the minimum amount that he can expect to pay for his premium? Hint: The minimum premium occurs when the insurance company’s expected profit is zero. 9. Suppose the probability that a person likes pizza is 0.72. (a) What are the odds that this person likes pizza. (b) What are the odds that this person does NOT like pizza? 4 c Math 141, Fall 2014, Dr. Kamran Reihani - Texas A&M University 10. There are three envelopes 1, 2 and 3. Envelope #1 has 4 red beads, 2 black beads, 3 green beads, and $5 in it. Envelope #2 has 2 red beads, 4 black beads, 1 green beads, and $2 in it. Envelope #3 has 1 red beads, 5 black beads, and no money in it. A fair die is rolled. If a six turns up, #1 is opened and a bead is selected at random from it. If a two or four turns up, #2 is opened and a bead is selected at random from it. If a one, three or five turns up, #3 is opened and a bead is selected at random from it. Suppose an envelope is selected at random as described above, but all you are shown in the end is a black bead. How much should you pay for the envelope? 11. In a horse race, a gambler wants to bet on a horse with a 1 to 9 odds of winning. He knows that if the horse wins, he will get an amount of money equal to 3 times his bet plus an extra $140. How much should he bet so that this “game” would be considered fair? 5 c Math 141, Fall 2014, Dr. Kamran Reihani - Texas A&M University 12. Determine whether the following experiments are binomial. (a) Picking 3 cards in succession with replacement from a deck of cards and observing if the card is a king. (b) Tossing a coin until a head is tossed. (c) Rolling a pair of dice 5 times and recording the sum. (d) Picking 4 marbles in succession without replacement from a jar of 3 red and 5 blue marbles and observing the color of the marble. 13. A fair die is rolled five times. Calculate the probabilities of obtaining: (a) Exactly two sixes. (b) At least two sixes. (c) At most two sixes. (d) Between two and four sixes (inclusive). (e) Find the expected number, standard deviation and variance of the number of sixes in this experiment. 6 c Math 141, Fall 2014, Dr. Kamran Reihani - Texas A&M University 14. An algebra quiz consists of eight multiple choice questions. Six must be answered correctly to receive a passing grade. If each question has five possible answers, of which only one is correct, what is the probability that a student who guesses at random on each question will pass the examination? 15. An automobile-manufacturing company uses ten industrial robots as welders on its assembly line. On a given working day, the probability that a robot will be operative is 0.95. What is the probability that on a given working day two to five robots will be inoperative? 7