Reliability For a parallel structure of identical

Problem 1E Two events E and F are______if the occurrence of event E in a probability experiment does not affect the probability of event F.

Problem 4E True or False: When two events are disjoint, they are also independent.

Problem 7E Determine whether the events E and F are independent or dependent. Justify your answer. (a) E: Speeding on the interstate. F: Being pulled over by a police officer. ________________ (b) E: You gain weight. F: You eat fast food for dinner every night. ________________ (c) E: You get a high score on a statistics exam. F: The Boston Red Sox win a baseball game.

Problem 3E The word or in probability implies that we use the ______ Rule.

Problem 6E Suppose events E and F are disjoint. What is P(E and F)?

Problem 2E The word and in probability implies that we use the______ Rule.

Problem 5E If two events E and F are independent, P(E and F) =______.

Problem 8E Determine whether the events E and F are independent or dependent. Justify your answer. (a) E: The battery in your cell phone is dead. F: The batteries in your calculator are dead. ________________ (b) E: Your favorite color is blue. ________________ F: Your friend’s favorite hobby is fishing. ________________ (c) E: You are late for school. F: Your car runs out of gas.

Problem 9E Suppose that events E and F are independent, P(E) = 0.3 and P(F) = 0.6. What is the P(E and F)?

Problem 11E Flipping a Coin What is the probability of obtaining five heads in a row when flipping a fair coin? Interpret this probability.

Problem 13E Southpaws About 13% of the population is left-handed. If two people are randomly selected, what is the probability that both are left-handed? What is the probability that at least one is right-handed?

Problem 14E Double Jackpot Shawn lives near the border of Illinois and Missouri. One weekend he decides to play $1 in both state lotteries in hopes of hitting two jackpots. The probability of winning the Missouri Lotto is about 0.00000028357 and the probability of winning the Illinois Lotto is about 0.000000098239. (a) Explain why the two lotteries are independent. ________________ (b) Find the probability that Shawn will win both jackpots.

Problem 10E Suppose that events E and F are independent, P(E) = 0.7 and P(F) = 0.9. What is the P(E and F)?

Problem 12E Rolling a Die What is the probability of obtaining 4 ones in a row when rolling a fair, six-sided die? Interpret this probability.

Problem 15E False Positives The ELISA is a test to determine whether the HIV antibody is present. The test is 99.5% effective, which means that the test will come back negative if the HIV antibody is not present 99.5% of the time. The probability of a test coming back positive when the antibody is not present (a false positive) is 0.005. Suppose that the ELISA is given to five randomly selected people who do not have the HIV antibody. (a) What is the probability that the ELISA comes back negative for all five people? ________________ (b) What is the probability that the ELISA comes back positive for at least one of the five people?

Problem 16E Christmas Lights Christmas lights are often designed with a series circuit. This means that when one light burns out the entire string of lights goes black. Suppose that the lights are designed so that the probability a bulb will last 2 years is 0.995. The success or failure of a bulb is independent of the success or failure of other bulbs. (a) What is the probability that in a string of 100 lights all 100 will last 2 years? ________________ (b) What is the probability that at least one bulb will burn out in 2 years?

Problem 19E Mental Illness According to the Department of Health and Human Services, 30% of 18- to 25-year-olds have some form of mental illness. (a) What is the probability two randomly selected 18- to 25-year-olds have some form of mental illness? ________________ (b) What is the probability six randomly selected 18- to 25-year-olds have some form of mental illness? ________________ (c) What is the probability at least one of six randomly selected 18- to 25-year-olds has some form of mental illness? ________________ (d) Would it be unusual that among four randomly selected 18-to 25-year-olds, none has some form of mental illness?

Problem 20E Quality Control Suppose that a company selects two people who work independently inspecting two-by-four timbers. Their job is to identify low-quality timbers. Suppose that the probability that an inspector does not identify a low-quality timber is 0.20. (a) What is the probability that both inspectors do not identify a low-quality timber? ________________ (b) How many inspectors should be hired to keep the probability of not identifying a low-quality timber below 1%? ________________ (c) Interpret the probability from part (a).

Problem 22E E.P.T. Pregnancy Tests The packaging of an E.P.T. Pregnancy Test states that the test is “99% accurate at detecting typical pregnancy hormone levels.” Assume that the probability that a test will correctly identify a pregnancy is 0.99 and that 12 randomly selected pregnant women with typical hormone levels are each given the test. (a) What is the probability that all 12 tests will be positive? ________________ (b) What is the probability that at least one test will not be positive?

Problem 17E Life Expectancy The probability that a randomly selected 40-year-old male will live to be 41 years old is 0.99757, according to the National Vital Statistics Report, Vol. 56, No. 9. (a) What is the probability that two randomly selected 40-year-old males will live to be 41 years old? ________________ (b) What is the probability that five randomly selected 40-year-old males will live to be 41 years old? ________________ (c) What is the probability that at least one of five randomly selected 40-year-old males will not live to be 41 years old? Would it be unusual if at least one of five randomly selected 40-year-old males did not live to be 41 years old?

Problem 21E Reliability For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.15 probability of failure. (a) Would it be unusual to observe one component fail? Two components? ________________ (b) What is the probability that a parallel structure with 2 identical components will succeed? ________________ (c) How many components would be needed in the structure so that the probability the system will succeed is greater than 0.9999?

Problem 18E Life Expectancy The probability that a randomly selected 40-year-old female will live to be 41 years old is 0.99855 according to the National Vital Statistics Report, Vol. 56, No. 9. (a) What is the probability that two randomly selected 40-year-old females will live to be 41 years old? ________________ (b) What is the probability that five randomly selected 40-year-old females will live to be 41 years old? ________________ (c) What is the probability that at least one of five randomly selected 40-year-old females will not live to be 41 years old? Would it be unusual if at least one of five randomly selected 40-year-old females did not live to be 41 years old?

Problem 23E Cold Streaks Players in sports are said to have “hot streaks” and “cold streaks.” For example, a batter in baseball might be considered to be in a slump, or cold streak, if he has made 10 outs in 10 consecutive at-bats. Suppose that a hitter successfully reaches base 30% of the time he comes to the plate. (a) Find and interpret the probability that the hitter makes 10 outs in 10 consecutive at-bats, assuming that at-bats are independent events. Hint: The hitter makes an out 70% of the time. ________________ (b) Are cold streaks unusual? ________________ (c) Find the probability the hitter makes five consecutive outs and then reaches base safely.

Problem 24E Hot Streaks In a recent basketball game, a player who makes 65% of his free throws made eight consecutive free throws. Assuming that free-throw shots are independent, determine whether this feat was unusual.

Problem 25E Bowling Suppose that Ralph gets a strike when bowling 30% of the time. (a) What is the probability that Ralph gets two strikes in a row? ________________ (b) What is the probability that Ralph gets a turkey (three strikes in a row)? ________________ (c) When events are independent, their complements are independent as well. Use this result to determine the probability that Ralph gets a turkey, but fails to get a clover (four strikes in a row).

Problem 26E NASCAR Fans Among Americans who consider themselves auto racing fans, 59% identify NASCAR stock cars as their favorite type of racing. Suppose that four auto racing fans are randomly selected. (a) What is the probability that all four will identify NASCAR stock cars as their favorite type of racing? ________________ (b) What is the probability that at least one will not identify NASCAR stock cars as his or her favorite type of racing? ________________ (c) What is the probability that none will identify NASCAR stock cars as his or her favorite type of racing? ________________ (d) What is the probability that at least one will identify NASCAR stock cars as his or her favorite type of racing?

Problem 28E Defense System Suppose that a satellite defense system is established in which four satellites acting independently have a 0.9 probability of detecting an incoming ballistic missile. What is the probability that at least one of the four satellites detects an incoming ballistic missile? Would you feel safe with such a system?

Problem 27E Driving under the Influence Among 21- to 25-year-olds, 29% say they have driven while under the influence of alcohol. Suppose that three 21- to 25-year-olds are selected at random. (a) What is the probability that all three have driven while under the influence of alcohol? ________________ (b) What is the probability that at least one has not driven while under the influence of alcohol? ________________ (c) What is the probability that none of the three has driven while under the influence of alcohol? ________________ (d) What is the probability that at least one has driven while under the influence of alcohol?

Problem 30E Weight Gain and Gender According to the National Vital Statistics Report, 20.1% of all pregnancies result in weight gain in excess of 40 pounds (for singleton births). In addition, 49.5% of all pregnancies result in the birth of a baby girl. Assuming gender and weight gain are independent, what is the probability a randomly selected pregnancy results in a girl and weight gain in excess of 40 pounds?

Problem 29E Audits and Pet Ownership According to Internal Revenue Service records, 6.42% of all household tax returns are audited. According to the Humane Society, 39% of all households own a dog. Assuming dog ownership and audits are independent events, what is the probability a randomly selected household is audited and owns a dog?

Problem 31E Stocks Suppose your financial advisor recommends three stocks to you. He claims the likelihood that the first stock will increase in value at least 10% within the next year is 0.7, the likelihood the second stock will increase in value at least 10% within the next year is 0.55, and the likelihood the third stock will increase at least 10% within the next year is 0.20. Would it be unusual for all three stocks to increase at least 10%, assuming the stocks behave independently of each other?

Problem 32E Betting on Sports According to a Gallup Poll, about 17% of adult Americans bet on professional sports. Census data indicate that 48.4% of the adult population in the United States is male. (a) Assuming that betting is independent of gender, compute the probability that an American adult selected at random is a male and bets on professional sports. ________________ (b) Using the result in part (a), compute the probability that an American adult selected at random is male or bets on professional sports. ________________ (c) The Gallup poll data indicated that 10.6% of adults in the United States are males and bet on professional sports. What does this indicate about the assumption in part (a)? ________________ (d) How will the information in part (c) affect the probability you computed in part (b)?

Birthdays Exclude leap years from the following calculations: 1. Compute the probability that a randomly selected person does not have a birthday on November 8. 2. Compute the probability that a randomly selected person does not have a birthday on the 1st day of a month. 3. Compute the probability that a randomly selected person does not have a birthday on the 31st day of a month. 4. Compute the probability that a randomly selected person was not born in December.

Two events E and F are _____ if the occurrence of event E in a probability experiment does not affect the probability of event F

The word and in probability implies that we use the _____ Rule.

The word or in probability implies that we use the _____ Rule

True or False: When two events are disjoint, they are also independent.

If two events E and F are independent, P(E and F) = _____ .

Suppose events E and F are disjoint. What is P(E and F )?

Determine whether the events E and F are independent or dependent. Justify your answer. (a) E: Speeding on the interstate. F: Being pulled over by a police officer. (b) E: You gain weight. F: You eat fast food for dinner every night. (c) E: You get a high score on a statistics exam. F: The Boston Red Sox win a baseball game

Determine whether the events E and F are independent or dependent. Justify your answer. (a) E: The battery in your cell phone is dead. F: The batteries in your calculator are dead. (b) E: Your favorite color is blue. F: Your friends favorite hobby is fishing. (c) E: You are late for school. F: Your car runs out of gas

Suppose that events E and F are independent, P(E) = 0.3 and P(F) = 0.6. What is the P(E and F )?

Suppose that events E and F are independent, P(E) = 0.7 and P(F) = 0.9. What is the P(E and F )?

Flipping a Coin What is the probability of obtaining five heads in a row when flipping a fair coin? Interpret this probability

Rolling a Die What is the probability of obtaining 4 ones in a row when rolling a fair, six-sided die? Interpret this probability

Southpaws About 13% of the population is left-handed. If two people are randomly selected, what is the probability that both are left-handed? What is the probability that at least one is right-handed?

Double Jackpot Shawn lives near the border of Illinois and Missouri. One weekend he decides to play $1 in both state lotteries in hopes of hitting two jackpots. The probability of winning the Missouri Lotto is about 0.00000028357 and the probability of winning the Illinois Lotto is about 0.000000098239. (a) Explain why the two lotteries are independent. (b) Find the probability that Shawn will win both jackpots.

False Positives The ELISA is a test to determine whether the HIV antibody is present. The test is 99.5% effective, which means that the test will come back negative if the HIV antibody is not present 99.5% of the time. The probability of a test coming back positive when the antibody is not present (a false positive) is 0.005. Suppose that the ELISA is given to five randomly selected people who do not have the HIV antibody. (a) What is the probability that the ELISA comes back negative for all five people? (b) What is the probability that the ELISA comes back positive for at least one of the five people?

Christmas Lights Christmas lights are often designed with a series circuit. This means that when one light burns out the entire string of lights goes black. Suppose that the lights are designed so that the probability a bulb will last 2 years is 0.995. The success or failure of a bulb is independent of the success or failure of other bulbs. (a) What is the probability that in a string of 100 lights all 100 will last 2 years? (b) What is the probability that at least one bulb will burn out in 2 years?

Life Expectancy The probability that a randomly selected 40-year-old male will live to be 41 years old is 0.99757, according to the National Vital Statistics Report, Vol. 56, No. 9. (a) What is the probability that two randomly selected 40-yearold males will live to be 41 years old? (b) What is the probability that five randomly selected 40-yearold males will live to be 41 years old? (c) What is the probability that at least one of five randomly selected 40-year-old males will not live to be 41 years old? Would it be unusual if at least one of five randomly selected 40-year-old males did not live to be 41 years old?

Life Expectancy The probability that a randomly selected 40-year-old female will live to be 41 years old is 0.99855 according to the National Vital Statistics Report, Vol. 56, No. 9. (a) What is the probability that two randomly selected 40-yearold females will live to be 41 years old? (b) What is the probability that five randomly selected 40-yearold females will live to be 41 years old? (c) What is the probability that at least one of five randomly selected 40-year-old females will not live to be 41 years old? Would it be unusual if at least one of five randomly selected 40-year-old females did not live to be 41 years old?

Mental Illness According to the Department of Health and Human Services, 30% of 18- to 25-year-olds have some form of mental illness. (a) What is the probability two randomly selected 18- to 25-yearolds have some form of mental illness? (b) What is the probability six randomly selected 18- to 25-yearolds have some form of mental illness? (c) What is the probability at least one of six randomly selected 18- to 25-year-olds has some form of mental illness? (d) Would it be unusual that among four randomly selected 18- to 25-year-olds, none has some form of mental illness?

Quality Control Suppose that a company selects two people who work independently inspecting two-by-four timbers. Their job is to identify low-quality timbers. Suppose that the probability that an inspector does not identify a low-quality timber is 0.20. (a) What is the probability that both inspectors do not identify a low-quality timber? (b) How many inspectors should be hired to keep the probability of not identifying a low-quality timber below 1%? (c) Interpret the probability from part (a).

Reliability For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.15 probability of failure. (a) Would it be unusual to observe one component fail? Two components? (b) What is the probability that a parallel structure with 2 identical components will succeed?(c) How many components would be needed in the structure so that the probability the system will succeed is greater than 0.9999?

E.P.T. Pregnancy Tests The packaging of an E.P.T. Pregnancy Test states that the test is 99% accurate at detecting typical pregnancy hormone levels. Assume that the probability that a test will correctly identify a pregnancy is 0.99 and that 12 randomly selected pregnant women with typical hormone levels are each given the test. (a) What is the probability that all 12 tests will be positive? (b) What is the probability that at least one test will not be positive?

Cold Streaks Players in sports are said to have hot streaks and cold streaks. For example, a batter in baseball might be considered to be in a slump, or cold streak, if he has made 10 outs in 10 consecutive at-bats. Suppose that a hitter successfully reaches base 30% of the time he comes to the plate. (a) Find and interpret the probability that the hitter makes 10 outs in 10 consecutive at-bats, assuming that at-bats are independent events. Hint: The hitter makes an out 70% of the time. (b) Are cold streaks unusual? (c) Find the probability the hitter makes five consecutive outs and then reaches base safely.

Hot Streaks In a recent basketball game, a player who makes 65% of his free throws made eight consecutive free throws. Assuming that free-throw shots are independent, determine whether this feat was unusual.

Bowling Suppose that Ralph gets a strike when bowling 30% of the time. (a) What is the probability that Ralph gets two strikes in a row? (b) What is the probability that Ralph gets a turkey (three strikes in a row)? (c) When events are independent, their complements are independent as well. Use this result to determine the probability that Ralph gets a turkey, but fails to get a clover (four strikes in a row).

NASCAR Fans Among Americans who consider themselves auto racing fans, 59% identify NASCAR stock cars as their favorite type of racing. Suppose that four auto racing fans are randomly selected. Source: ESPN/TNS Sports, reported in USA Today (a) What is the probability that all four will identify NASCAR stock cars as their favorite type of racing? (b) What is the probability that at least one will not identify NASCAR stock cars as his or her favorite type of racing? (c) What is the probability that none will identify NASCAR stock cars as his or her favorite type of racing? (d) What is the probability that at least one will identify NASCAR stock cars as his or her favorite type of racing?

Driving under the Influence Among 21- to 25-year-olds, 29% say they have driven while under the influence of alcohol. Suppose that three 21- to 25-year-olds are selected at random. Source: U.S. Department of Health and Human Services, reported in USA Today (a) What is the probability that all three have driven while under the influence of alcohol? (b) What is the probability that at least one has not driven while under the influence of alcohol? (c) What is the probability that none of the three has driven while under the influence of alcohol? (d) What is the probability that at least one has driven while under the influence of alcohol?

Defense System Suppose that a satellite defense system is established in which four satellites acting independently have a 0.9 probability of detecting an incoming ballistic missile. What is the probability that at least one of the four satellites detects an incoming ballistic missile? Would you feel safe with such a system?

Audits and Pet Ownership According to Internal Revenue Service records, 6.42% of all household tax returns are audited. According to the Humane Society, 39% of all households own a dog. Assuming dog ownership and audits are independent events, what is the probability a randomly selected household is audited and owns a dog?

Weight Gain and Gender According to the National Vital Statistics Report, 20.1% of all pregnancies result in weight gain in excess of 40 pounds (for singleton births). In addition, 49.5% of all pregnancies result in the birth of a baby girl. Assuming gender and weight gain are independent, what is the probability a randomly selected pregnancy results in a girl and weight gain in excess of 40 pounds?

Stocks Suppose your financial advisor recommends three stocks to you. He claims the likelihood that the first stock will increase in value at least 10% within the next year is 0.7, the likelihood the second stock will increase in value at least 10% within the next year is 0.55, and the likelihood the third stock will increase at least 10% within the next year is 0.20. Would it be unusual for all three stocks to increase at least 10%, assuming the stocks behave independently of each other?

Betting on Sports According to a Gallup Poll, about 17% of adult Americans bet on professional sports. Census data indicate that 48.4% of the adult population in the United States is male.(a) Assuming that betting is independent of gender, compute the probability that an American adult selected at random is a male and bets on professional sports. (b) Using the result in part (a), compute the probability that an American adult selected at random is male or bets on professional sports. (c) The Gallup poll data indicated that 10.6% of adults in the United States are males and bet on professional sports. What does this indicate about the assumption in part (a)? (d) How will the information in part (c) affect the probability you computed in part (b)?

Fingerprints Fingerprints are now widely accepted as a form of identification. In fact, many computers today use fingerprint identification to link the owner to the computer. In 1892, Sir Francis Galton explored the use of fingerprints to uniquely identify an individual. A fingerprint consists of ridgelines. Based on empirical evidence, Galton estimated the probability that a square consisting of six ridgelines that covered a fingerprint could be filled in accurately by an experienced fingerprint analyst as 1 2 . (a) Assuming that a full fingerprint consists of 24 of these squares, what is the probability that all 24 squares could be filled in correctly, assuming that success or failure in filling in one square is independent of success or failure in filling in any other square within the region? (This value represents the probability that two individuals would share the same ridgeline features within the 24-square region.) (b) Galton further estimated that the likelihood of determining the fingerprint type (e.g., arch, left loop, whorl, etc.) as a 1 2 b 4 and the likelihood of the occurrence of the correct number of ridges entering and exiting each of the 24 regions as a 1 2 b 8 . Assuming that all three probabilities are independent, compute Galtons estimate of the probability that a particular fingerprint configuration would occur in nature (that is, the probability that a fingerprint match occurs by chance).