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7. A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. a) What is the probability that it is the fair coin? b) Suppose that he flips the same coin a second time and again it shows heads. Now what is the probability that it is the fair coin? c) Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin?

7. A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. a) What is the probability that it is the fair coin? b) Suppose that he flips the same coin a second time and again it shows heads. Now what is the probability that it is the fair coin? c) Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Please answer 7 and 8

2. Suppose that 5 percent of men and 0.25 percent of women are color-blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females. 3. Two cards are randomly selected from a deck of 52 playing cards. a) What is the probability they constitute a pair (that is, that they are of the same denomination)? b) What is the conditional probability they constitute a pair given that they are of different suits? 4. Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability 0.7, whereas George, independently, hits the target with probability 0.4. a) Given that exactly one shot hit the target, what is the probability that it was George's shot? b) Given that the target is hit, what is the probability that George hit it? 5. A fair coin is continually flipped. What is the probability that the first four flips are а) Н, Н, Н, Н? b) T, Н, Н, Н? c) What is the probability that the pattern T, H, H, H occurs before the pattern Н, Н, Н, Н? 6. Urn 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in urn 2. A ball is then drawn from urn 2. It happens to be white. What is the probability that the transferred ball was white? 2/3 7. A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. a) What is the probability that it is the fair coin? b) Suppose that he flips the same coin a second time and again it shows heads. Now what is the probability that it is the fair coin? c) Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin? 8. Entomological engineers are continually searching for new biological agents to control one of the world's worst aquatic weeds, the water hyacinth. An insect that naturally feeds on water hyacinth is the delphacid. Female delphacids lay anywhere from one to four eggs onto a water hyacinth blade. The Annals of the Entomological Society of America (Jan. 2005) published a study of the life cycle of a South American delphacid species. The accompanying table gives the percentages of water hyacinth blades that have one, two, three, and four delphacid eggs. One Egg Two Egg Three Egg Four Egg Percentage of Blades 40 54 4 Source: Sosa, A. J., et al. “Life history of Megamelus scutellaris with description of immature stages," Annals of the Entomological Society of America, Vol. 98, No. 1, Jan. 2005 (adapted from Table 1). a) One of the water hyacinth blades in the study is randomly selected and Y, the number of delphacid eggs on the blade, is observed. Give the probability distribution of Y. b) What is the probability that the blade has at least three delphacid eggs? 9. A weapons manufacturer uses a liquid propellant to produce gun cartridges. During the manufacturing process, the propellant can get mixed with another liquid to produce a contaminated cartridge. A University of South Florida statistician, hired by the company to investigate the level of contamination in the stored cartridges, found that 23% of the cartridges in a particular lot were contaminated. Suppose you randomly sample (without replacement) gun cartridges from this lot until you find a contaminated one. Let Y= y be the number of cartridges sampled until a contaminated one is found. It is known that the probability distribution for is given by the formula: p(y)=(0.23)(0.77)", y = 1,2,3,... a) Find p(1). Interpret this result. b) Find p(5). Interpret this result.
Transcribed Image Text:2. Suppose that 5 percent of men and 0.25 percent of women are color-blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females. 3. Two cards are randomly selected from a deck of 52 playing cards. a) What is the probability they constitute a pair (that is, that they are of the same denomination)? b) What is the conditional probability they constitute a pair given that they are of different suits? 4. Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability 0.7, whereas George, independently, hits the target with probability 0.4. a) Given that exactly one shot hit the target, what is the probability that it was George's shot? b) Given that the target is hit, what is the probability that George hit it? 5. A fair coin is continually flipped. What is the probability that the first four flips are а) Н, Н, Н, Н? b) T, Н, Н, Н? c) What is the probability that the pattern T, H, H, H occurs before the pattern Н, Н, Н, Н? 6. Urn 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in urn 2. A ball is then drawn from urn 2. It happens to be white. What is the probability that the transferred ball was white? 2/3 7. A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. a) What is the probability that it is the fair coin? b) Suppose that he flips the same coin a second time and again it shows heads. Now what is the probability that it is the fair coin? c) Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin? 8. Entomological engineers are continually searching for new biological agents to control one of the world's worst aquatic weeds, the water hyacinth. An insect that naturally feeds on water hyacinth is the delphacid. Female delphacids lay anywhere from one to four eggs onto a water hyacinth blade. The Annals of the Entomological Society of America (Jan. 2005) published a study of the life cycle of a South American delphacid species. The accompanying table gives the percentages of water hyacinth blades that have one, two, three, and four delphacid eggs. One Egg Two Egg Three Egg Four Egg Percentage of Blades 40 54 4 Source: Sosa, A. J., et al. “Life history of Megamelus scutellaris with description of immature stages," Annals of the Entomological Society of America, Vol. 98, No. 1, Jan. 2005 (adapted from Table 1). a) One of the water hyacinth blades in the study is randomly selected and Y, the number of delphacid eggs on the blade, is observed. Give the probability distribution of Y. b) What is the probability that the blade has at least three delphacid eggs? 9. A weapons manufacturer uses a liquid propellant to produce gun cartridges. During the manufacturing process, the propellant can get mixed with another liquid to produce a contaminated cartridge. A University of South Florida statistician, hired by the company to investigate the level of contamination in the stored cartridges, found that 23% of the cartridges in a particular lot were contaminated. Suppose you randomly sample (without replacement) gun cartridges from this lot until you find a contaminated one. Let Y= y be the number of cartridges sampled until a contaminated one is found. It is known that the probability distribution for is given by the formula: p(y)=(0.23)(0.77)", y = 1,2,3,... a) Find p(1). Interpret this result. b) Find p(5). Interpret this result.
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