Solution: Identical and Fraternal Twins. In Exercises 2528, use the data in the

Subjective Probability a. Estimate the probability that on the next test in a randomly selected statistics class, at least one student earns a grade of A. b. Answer part (a) with the additional knowledge that the selected class is a special section for honors students with very high grade point averages.

Conditional Probability What is a conditional probability?

Notation Let event A = subject is telling the truth and event B = polygraph test indicates that the subject is lying. Use your own words to translate the notation P ( B | A ) into a verbal statement.

Confusion of the Inverse Using the same events A and B described in Exercise 3, what is confusion of the inverse? (Express the answer in both words and symbols.)

Describing Complements. In Exercises 58, provide a written description of the complement of the given event, then find the probability of the complement of the given event. Five Girls When a couple has five children, all five are girls. (Assume that boys and girls are equally likely.)

Describing Complements. In Exercises 58, provide a written description of the complement of the given event, then find the probability of the complement of the given event.No Girls When a couple has five children, none of the five is a girl. (Assume that boys and girls are equally likely.)

Describing Complements. In Exercises 58, provide a written description of the complement of the given event, then find the probability of the complement of the given event.At Least One Zero When four digits (between 0 and 9 inclusive) are randomly selected with replacement for a lottery ticket, at least one of the digits is 0.

Describing Complements. In Exercises 58, provide a written description of the complement of the given event, then find the probability of the complement of the given event. No 7s When four digits (between 0 and 9 inclusive) are randomly selected with replacement for a lottery ticket, none of the digits is a 7.

At Least One Correct Answer If you make random guesses for 10 multiplechoice test questions (each with five possible answers), what is the probability of getting at least 1 correct? If a very lenient instructor says that passing the test occurs if there is at least one correct answer, can you reasonably expect to pass by guessing?

At Least One Working Calculator A statistics student plans to use a TI84 Plus calculator on her final exam. From past experience, she estimates that there is 0.92 probability that the calculator will work on any given day. Because the final exam is so important, she plans to use redundancy by bringing in two TI84 Plus calculators. What is the probability that she will be able to complete her exam with a working calculator? Does she really gain much by bringing in the backup calculator? Explain.

Probability of a Girl Assuming that boys and girls are equally likely, find the probability of a couple having at least one boy after their fifth child is born, given that the first four children were all girls.

At Least One Correct Answer If you make random guesses for 10 multiplechoice test questions (each with five possible answers), what is the probability of getting at least one answer correct, given that the first nine answers are all wrong?

Births in the United States In the United States, the true probability of a baby being a boy is 0.512 (based on the data available at this writing). Among the next five randomly selected births in the United States, what is the probability that at least one of them is a girl?

Births in China In China, the probability of a baby being a boy is 0.545. Many couples are allowed to have only one child. Among the next five randomly selected births in China, what is the probability that at least one of them is a girl? Can this system continue to work indefinitely?

Car Crashes The probability of a randomly selected car crashing during a year is 0.0423 (based on data from the Statistical Abstract of the United States). If a family has three cars, find the probability that at least one of them has a crash during the year. Is there any reason why this probability might be wrong?

Cleared Burglaries According to FBI data, 12.4% of burglaries are cleared with arrests. A new detective is assigned to five different burglaries. a. What is the probability that at least one of them is cleared with an arrest? b. What is the probability that the detective clears five burglaries with arrests? c. What should we conclude if the detective clears all five burglaries with arrests?

WiFi Based on a poll conducted through the eedition of USA Today, 67% of Internet users are more careful about personal information when using a public WiFi hotspot. What is the probability that among four randomly selected Internet users, at least one is more careful about personal information when using a public WiFi hotspot? How is the result affected by the additional information that the survey subjects volunteered to respond?

Compliments at Work Based on a poll conducted through email by USA Today, 41% of survey respondents most liked to get compliments at work from their coworkers. Among 12 randomly selected workers, what is the probability of getting at least 1 who most likes to get compliments from coworkers? How is the result affected by the additional information that the survey subjects volunteered to respond?

In Exercises 1924, refer to Table 41, included with the Chapter Problem. In each case, assume that 1 of the 1000 test subjects is randomly selected False Positive Find the probability of selecting a subject with a positive test result, given that the subject does not use drugs. Why is this particular case problematic for test subjects?

In Exercises 1924, refer to Table 41, included with the Chapter Problem. In each case, assume that 1 of the 1000 test subjects is randomly selectedFalse Negative Find the probability of selecting a subject with a negative test result, given that the subject uses drugs. Who would suffer from this type of error?

In Exercises 1924, refer to Table 41, included with the Chapter Problem. In each case, assume that 1 of the 1000 test subjects is randomly selectedInverse Probabilities Find P ( subject uses drugs | negative test result ) . Compare this result to the result found in Exercise 20. Are P ( subject uses drugs | negative test result ) . and P ( negative test result | subject uses drugs ) equal?

In Exercises 1924, refer to Table 41, included with the Chapter Problem. In each case, assume that 1 of the 1000 test subjects is randomly selected. Inverse Probabilities a. Find P ( negative test result | subject uses drugs ) . b. Find P ( subject uses drugs | negative test result ) . c. Compare the results from parts (a) and (b). Are they equal?

In Exercises 1924, refer to Table 41, included with the Chapter Problem. In each case, assume that 1 of the 1000 test subjects is randomly selectedPositive Predictive Value Find the positive predictive value for the test. That is, find the probability that a subject uses drugs, given that the test yields a positive result.

In Exercises 1924, refer to Table 41, included with the Chapter Problem. In each case, assume that 1 of the 1000 test subjects is randomly selectedNegative Predictive Value Find the negative predictive value for the test. That is, find the probability that a subject does not use drugs, given that the test yields a negative result.

Identical and Fraternal Twins. In Exercises 2528, use the data in the following table. Instead of summarizing observed results, the entries reflect the actual probabilities based on births of twins (based on data from the Northern California Twin Registry and the article Bayesians, Frequentists, and Scientists by Bradley Efron, Journal of the American Statistical Association, Vol. 100, No. 469). Identical twins come from a single egg that splits into two embryos, and fraternal twins are from separate fertilized eggs.The table entries reflect the principle that among sets of twins, 1/3 are identical and 2/3 are fraternal. Also, identical twins must be of the same sex and the sexes are equally likely (approximately), and sexes of fraternal twins are equally likely. Sexes of Twins boy/boy boy/girl girl/boy girl/girl Identical Twins 5 0 0 5 Fraternal Twins 5 5 5 5 Identical Twins a. After having a sonogram, a pregnant woman learns that she will have twins. What is the probability that she will have identical twins? b. After studying the sonogram more closely, the physician tells the pregnant woman that she will give birth to twin boys. What is the probability that she will have identical twins? That is, find the probability of identical twins given that the twins consist of two boys.

Identical and Fraternal Twins. In Exercises 2528, use the data in the following table. Instead of summarizing observed results, the entries reflect the actual probabilities based on births of twins (based on data from the Northern California Twin Registry and the article Bayesians, Frequentists, and Scientists by Bradley Efron, Journal of the American Statistical Association, Vol. 100, No. 469). Identical twins come from a single egg that splits into two embryos, and fraternal twins are from separate fertilized eggs.The table entries reflect the principle that among sets of twins, 1/3 are identical and 2/3 are fraternal. Also, identical twins must be of the same sex and the sexes are equally likely (approximately), and sexes of fraternal twins are equally likely. Sexes of Twins boy/boy boy/girl girl/boy girl/girl Identical Twins 5 0 0 5 Fraternal Twins 5 5 5 5 Fraternal Twins a. After having a sonogram, a pregnant woman learns that she will have twins. What is the probability that she will have fraternal twins? b. After studying the sonogram more closely, the physician tells the pregnant woman that she will give birth to twins consisting of one boy and one girl. What is the probability that she will have fraternal twins?

Identical and Fraternal Twins. In Exercises 2528, use the data in the following table. Instead of summarizing observed results, the entries reflect the actual probabilities based on births of twins (based on data from the Northern California Twin Registry and the article Bayesians, Frequentists, and Scientists by Bradley Efron, Journal of the American Statistical Association, Vol. 100, No. 469). Identical twins come from a single egg that splits into two embryos, and fraternal twins are from separate fertilized eggs.The table entries reflect the principle that among sets of twins, 1/3 are identical and 2/3 are fraternal. Also, identical twins must be of the same sex and the sexes are equally likely (approximately), and sexes of fraternal twins are equally likely. Sexes of Twins boy/boy boy/girl girl/boy girl/girl Identical Twins 5 0 0 5 Fraternal Twins 5 5 5 5 Fraternal Twins If a pregnant woman is told that she will give birth to fraternal twins, what is the probability that she will have one child of each sex?

Identical and Fraternal Twins. In Exercises 2528, use the data in the following table. Instead of summarizing observed results, the entries reflect the actual probabilities based on births of twins (based on data from the Northern California Twin Registry and the article Bayesians, Frequentists, and Scientists by Bradley Efron, Journal of the American Statistical Association, Vol. 100, No. 469). Identical twins come from a single egg that splits into two embryos, and fraternal twins are from separate fertilized eggs.The table entries reflect the principle that among sets of twins, 1/3 are identical and 2/3 are fraternal. Also, identical twins must be of the same sex and the sexes are equally likely (approximately), and sexes of fraternal twins are equally likely. Sexes of Twins boy/boy boy/girl girl/boy girl/girl Identical Twins 5 0 0 5 Fraternal Twins 5 5 5 5 Fraternal Twins If a pregnant woman is told that she will give birth to fraternal twins, what is the probability that she will give birth to two girls?

Redundancy in Computer Hard Drives Assume that there is a 2% rate of disk drive failure in a year (based on data from various sources, including Failure Trends in Large Disk Drive Population, by Pinhero et al. of Google, Inc.). a. If all of your computer data is stored on a hard disk drive with a copy stored on a second hard disk drive, what is the probability that during a year, you can avoid catastrophe with at least one working drive? b. If copies of all of your computer data are stored on three independent hard disk drives, what is the probability that during a year, you can avoid catastrophe with at least one working drive?

Redundancy in Aircraft Radios The FAA requires that commercial aircraft used for flying in instrument conditions must have two independent radios instead of one. Assume that for a typical flight, the probability of a radio failure is 0.0035. What is the probability that a particular flight will be safe with at least one working radio? Why does the usual rounding rule of three significant digits not work here? Is this probability high enough to ensure flight safety?

Composite Drug Screening Based on the data in Table 41, assume that the probability of a randomly selected person testing positive for drug use is 0.134. If drug screening samples are collected from 8 random subjects and combined, find the probability that the combined sample will reveal a positive result. Is that probability low enough so that further testing of the individual samples is rarely necessary?

Composite Water Samples The Fairfield County Department of Public Health tests water for contamination due to the presence of E. coli (Escherichia coli) bacteria. To reduce laboratory costs, water samples from five public swimming areas are combined for one test, and further testing is done only if the combined sample tests positive.Based on past results, there is a 0.005 probability of finding E. coli bacteria in a public swimming area. Find the probability that a combined sample from five public swimming areas will reveal the presence of E. coli bacteria. Is that probability low enough so that further testing of the individual samples is rarely necessary?

Shared Birthdays Find the probability that of 25 randomly selected people, a. no 2 share the same birthday. b. at least 2 share the same birthday.

Unseen Coins A statistics professor tosses two coins that cannot be seen by any students. One student asks this question: Did one of the coins turn up heads? Given that the professor's response is yes, find the probability that both coins turned up heads.

Confusion of the Inverse In one study, physicians were asked to estimate the probability of a malignant cancer given that a test showed a positive result. They were told that the cancer had a prevalance rate of 1%, the test has a false positive rate of 10%, and the test is 80% accurate in correctly identifying a malignancy when the subject actually has the cancer. (See Probabilistic Reasoning in Clinical Medicine by David Eddy, Cambridge University Press.) a. Find P ( malignant | positive test result ) . (Hint: Assume that the study involves 1000 subjects and use the given information to construct a table with the same format as Table 41.) b. Find P ( positive test result | malignant ) . (Hint: Assume that the study involves 1000 subjects and construct a table with the same format as Table 4 1.) c. Out of 100 physicians, 95 estimated P ( malignant | positive test result ) . to be about 75%. Were those estimates reasonably accurate, or did they exhibit confusion of the inverse? What would be a consequence of confusion of the inverse in this situation?