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Math 141 WIR8_Dr. Rosanna Pearlstein 1. Urn A contains 2 white and 4 red balls. Urn B contains 1 white and 1 red ball. A ball is first drawn from urn A, and then put in urn B. A second ball is then drawn from urn B. a. What is the probability that the ball drawn from urn B is white? b. What is the (conditional) probability that the transferred ball was white given that a white ball is drawn from urn B? 2. A pair of fair six-sided dice is rolled. What is the probability that a. The sum of the uppermost numbers is less than or equal to 6? b. At least one 1 is cast? 3. The sample space 𝑆 = {𝑠1 , 𝑠2 , 𝑠3 } has the property that 𝑃(𝑠1 ) + 𝑃(𝑠2 ) = 0.47 and 𝑃(𝑠2 ) + 𝑃(𝑠3 ) = 0.72. Find the probability of each outcome 𝑠1 , 𝑠2 , 𝑠3 . 4. An experiment consists of drawing a card, then another, at random, from a well-shuffled standard deck of 52 playing cards. Find the probability that: a. the second card drawn is a queen. b. the first card was a queen given that the second card is a queen. 5. Let E and F be two events of an experiment with sample space S. Compute the probabilities of the events 𝐸 𝑐 , 𝐸 ∩ 𝐹, 𝐸 ∪ 𝐹, 𝐸 𝑐 ∩ 𝐹 𝑐 , 𝐸 𝑐 ∪ 𝐹 𝑐 , 𝐸 ∪ 𝐹 𝑐 if: a. 𝑃(𝐸) = 0.51, 𝑃(𝐹) = 0.32 , 𝑃(𝐸 𝑐 ∩ 𝐹) = 0.14 . b. The events E and F are mutually exclusive with 𝑃(𝐸) = 0.2, 𝑃(𝐹) = 0.7. 6. In the tree diagram below, compute the probabilities: 𝑃(𝐸), 𝑃(𝐹), 𝑃(𝐴|𝐸), 𝑃(𝐴|𝐹), 𝑃(𝐵|𝐸), 𝑃(𝐵|𝐹), 𝑃(𝐴), 𝑃(𝐵), 𝑃(𝐸|𝐴), 𝑃(𝐸|𝐵), 𝑃(𝐹|𝐴), 𝑃(𝐹|𝐵). A 0.15 E 0.8 0.85 B . 0.2 A 0.65 F 0.35 B 7. (From Durbin et.al. "Biological Sequence Analysis", Cambridge University Press, 1998) A rare genetic disease is discovered. Although only one in a million people carry it, you consider getting screened. You are told that the genetic test is extremely good; it is 100% sensitive (it is always correct if you have the disease) and 99.99% specific (it gives a false positive result only 0.01% of the time). Having recently learned Bayes' theorem, you decide not to take the test. Why? 8. On the basis of data obtained from the National Institute of dental Research, it has been determined that 42% of 12-year-olds have never had a cavity, 34% of 13-year-olds have never had a cavity, and 28% of 14-year-olds have never had a cavity. Suppose a child is selected at random from a group of 24 junior high school students that includes six 12-year-olds, eight 13-year-olds, and ten 14-year-olds. If this child does not have a cavity, what is the probability that this child is 14 years old? 9. A survey conducted by an independent agency for the National Lung Society found that of 2000 women, 680 were heavy smokers and 50 had emphysema. Of those who had emphysema, 42 were also heavy smokers. Using the data in this survey, determine whether the events “being a heavy smoker” and “having emphysema” are independent events. 10. The Acrosonic model F loudspeaker system has four loudspeaker components: a woofer, a midrange, a tweeter, and an electrical crossover. The quality-control manager of Acrosonic has determined that on the average 1% of the woofers, 0.8% of the midranges, and .5% of the tweeters are defective, while 1.5% of the electrical crossovers are defective. Determine the probability that a loudspeaker system selected at random coming off the assembly line and before final inspection is not defective. Assume that the defects in the manufacturing of the components are unrelated. 11. A company surveyed 1000 people on their age and the number of jeans purchased annually. The results of the poll are shown in the table below. Jeans Purchased 0 Annually Under 12 0 12-18 21 19-25 40 Over 25 73 Total 134 3 or more Total 77 70 63 50 147 52 54 126 40 64 96 27 245 439 182 210 270 260 260 1000 1 2 Find the probability that: a) The person is over 25 and purchases 3 or more jeans annually. b) The person is older than 18 or purchases 1 pair of jeans annually. c) The person is in the age group 12-18 and purchases at most 2 pairs of jeans annually.