Suppose that in Exercise 2.4-1, X = 1 if a red ball is

Problem 18E In group testing for a certain disease, a blood sample was taken from each of n individuals and part of each sample was placed in a common pool. The latter was then tested. If the result was negative, there was no more testing and all n individuals were declared negative with one test. If, however, the combined result was found positive, all individuals were tested, requiring n+1 tests. If p = 0.05 is the probability of a person’s having the disease and n = 5, compute the expected number of tests needed, assuming independence.

Problem 1E An urn contains 7 red and 11 white balls. Draw one ball at random from the urn. Let X = 1 if a red ball is drawn, and let X = 0 if a white ball is drawn. Give the pmf, mean, and variance of X.

Problem 2E Suppose that in Exercise 2.4-1, X = 1 if a red ball is drawn and X = ?1 if a white ball is drawn. Give the pmf, mean, and variance of X. References Exercise 2.4-1 An urn contains 7 red and 11 white balls. Draw one ball at random from the urn. Let X = 1 if a red ball is drawn, and let X = 0 if a white ball is drawn. Give the pmf, mean, and variance of X.

Problem 5E In a lab experiment involving inorganic syntheses of molecular precursors to organometallic ceramics, the final step of a five-step reaction involves the formation of a metal–metal bond. The probability of such a bond forming is p = 0.20. Let X equal the number of successful reactions out of n = 25 such experiments. (a) Find the probability that X is at most 4. (b) Find the probability that X is at least 5. (c) Find the probability that X is equal to 6. (d) Give the mean, variance, and standard deviation of X.

Problem 4E It is claimed that 15% of the ducks in a particular region have patent schistosome infection. Suppose that seven ducks are selected at random. Let X equal the number of ducks that are infected. (a) Assuming independence, how is X distributed? (b) Find (i) P(X ? 2), (ii) P(X = 1), and (iii) P(X ? 3).

Problem 3E On a six-question multiple-choice test there are five possible answers for each question, of which one is correct (C) and four are incorrect (I). If a student guesses randomly and independently, find the probability of (a) Being correct only on questions 1 and 4 (i.e., scoring C, I, I, C, I, I). (b) Being correct on two questions.

Problem 7E Suppose that 2000 points are selected independently and at random from the unit square {(x, y) : 0 ? x < 1, 0 ? y < 1}. Let W equal the number of points that fall into A = {(x, y) : x2 + y2 < 1}. (a) How is W distributed? (b) Give the mean, variance, and standard deviation ofW. (c) What is the expected value of W/500? (d) Use the computer to select 2000 pairs of random numbers. Determine the value of W and use that value to find an estimate for ?. (Of course, we know the real value of ?, and more will be said about estimation later in this text.) (e) How could you extend part (d) to estimate the volume V = (4/3)? of a ball of radius 1 in 3-space? (f) How could you extend these techniques to estimate the “volume” of a ball of radius 1 in n-space?

Problem 8E A boiler has four relief valves. The probability that each opens properly is 0.99. (a) Find the probability that at least one opens properly. (b) Find the probability that all four open properly.

Problem 9E Suppose that the percentage of American drivers who are multitaskers (e.g., talk on cell phones, eat a snack, or text message at the same time they are driving) is approximately 80%. In a random sample of n = 20 drivers, let X equal the number of multitaskers. (a) How is X distributed? (b) Give the values of the mean, variance, and standard deviation of X. (c) Find the following probabilities: (i) P(X = 15), (ii) P(X > 15), and (iii) P(X ? 15).

Problem 13E It is claimed that for a particular lottery, 1/10 of the 50 million tickets will win a prize. What is the probability of winning at least one prize if you purchase (a) 10 tickets or (b) 15 tickets?

Problem 10E A certain type of mint has a label weight of 20.4 grams. Suppose that the probability is 0.90 that a mint weighs more than 20.7 grams. Let X equal the number of mints that weigh more than 20.7 grams in a sample of eight mints selected at random. (a) How is X distributed if we assume independence? (b) Find (i) P(X = 8), (ii) P(X ? 6), and (iii) P(X ? 6).

Problem 11E A random variable X has a binomial distribution with mean 6 and variance 3.6. Find P(X = 4).

Problem 14E For the lottery described in Exercise 2.4-13, find the smallest number of tickets that must be purchased so that the probability of winning at least one prize is greater than (a) 0.50; (b) 0.95. Reference Exercise 2.4-13 It is claimed that for a particular lottery, 1/10 of the 50 million tickets will win a prize. What is the probability of winning at least one prize if you purchase (a) 10 tickets or (b) 15 tickets?

Problem 15E A hospital obtains 40% of its flu vaccine from Company A, 50% from Company B, and 10% from Company C. From past experience, it is known that 3% of the vials from A are ineffective, 2%from B are ineffective, and 5% from C are ineffective. The hospital tests five vials from each shipment. If at least one of the five is ineffective, find the conditional probability of that shipment’s having come from C.

Problem 16E A company starts a fund of M dollars from which it pays $1000 to each employee who achieves high performance during the year. The probability of each employee achieving this goal is 0.10 and is independent of the probabilities of the other employees doing so. If there are n = 10 employees, how much should M equal so that the fund has a probability of at least 99% of covering those payments?

Problem 17E Your stockbroker is free to take your calls about 60% of the time; otherwise, he is talking to another client or is out of the office. You call him at five random times during a given month. (Assume independence.) (a) What is the probability that he will take every one of the five calls? (b) What is the probability that he will accept exactly three of your five calls? (c) What is the probability that he will accept at least one of the calls?

An urn contains 7 red and 11 white balls. Draw one ball at random from the urn. Let X = 1 if a red ball is drawn, and let X = 0 if a white ball is drawn. Give the pmf, mean, and variance of X.

Suppose that in Exercise 2.4-1, X = 1 if a red ball is drawn and X = 1 if a white ball is drawn. Give the pmf, mean, and variance of X.

On a six-question multiple-choice test there are five possible answers for each question, of which one is correct (C) and four are incorrect (I). If a student guesses randomly and independently, find the probability of (a) Being correct only on questions 1 and 4 (i.e., scoring C, I, I, C, I, I). (b) Being correct on two questions.

It is claimed that 15% of the ducks in a particular region have patent schistosome infection. Suppose that seven ducks are selected at random. Let X equal the number of ducks that are infected. (a) Assuming independence, how is X distributed? (b) Find (i) \(P(X \geq 2)\), (ii) P(X = 1), and (iii) \(P(X \leq 3)\).

In a lab experiment involving inorganic syntheses of molecular precursors to organometallic ceramics, the final step of a five-step reaction involves the formation of a metalmetal bond. The probability of such a bond forming is p = 0.20. Let X equal the number of successful reactions out of n = 25 such experiments. (a) Find the probability that X is at most 4. (b) Find the probability that X is at least 5. (c) Find the probability that X is equal to 6. (d) Give the mean, variance, and standard deviation of X

It is believed that approximately 75% of American youth now have insurance due to the health care law. Suppose this is true, and let X equal the number of American youth in a random sample of n = 15 with private health insurance. (a) How is X distributed? (b) Find the probability that X is at least 10. (c) Find the probability that X is at most 10. (d) Find the probability that X is equal to 10. (e) Give the mean, variance, and standard deviation of X.

Suppose that 2000 points are selected independently and at random from the unit square {(x, y): 0 x < 1, 0 y < 1}. Let W equal the number of points that fall into A = {(x, y): x2 + y2 < 1}(a) How is W distributed? (b) Give the mean, variance, and standard deviation of W. (c) What is the expected value of W/500? (d) Use the computer to select 2000 pairs of random numbers. Determine the value of W and use that value to find an estimate for . (Of course, we know the real value of , and more will be said about estimation later in this text.) (e) How could you extend part (d) to estimate the volume V = (4/3) of a ball of radius 1 in 3-space? (f) How could you extend these techniques to estimate the volume of a ball of radius 1 in n-space?

A boiler has four relief valves. The probability that each opens properly is 0.99. (a) Find the probability that at least one opens properly. (b) Find the probability that all four open properly.

Suppose that the percentage of American drivers who are multitaskers (e.g., talk on cell phones, eat a snack, or text message at the same time they are driving) is approximately 80%. In a random sample of n = 20 drivers, let X equal the number of multitaskers. (a) How is X distributed? (b) Give the values of the mean, variance, and standard deviation of X. (c) Find the following probabilities: (i) P(X = 15), (ii) P(X > 15), and (iii) P(X 15).

A certain type of mint has a label weight of 20.4 grams. Suppose that the probability is 0.90 that a mint weighs more than 20.7 grams. Let X equal the number of mints that weigh more than 20.7 grams in a sample of eight mints selected at random. (a) How is X distributed if we assume independence? (b) Find (i) P(X = 8), (ii) P(X 6), and (iii) P(X 6).

A random variable X has a binomial distribution with mean 6 and variance 3.6. Find P(X = 4).

In the casino game chuck-a-luck, three fair sixsided dice are rolled. One possible bet is $1 on fives, and the payoff is equal to $1 for each five on that roll. In addition, the dollar bet is returned if at least one five is rolled. The dollar that was bet is lost only if no fives are rolled. Let X denote the payoff for this game. Then X can equal l, l, 2, or 3. (a) Determine the pmf f(x). (b) Calculate , 2, and . (c) Depict the pmf as a probability histogram.

It is claimed that for a particular lottery, 1/10 of the 50 million tickets will win a prize. What is the probability of winning at least one prize if you purchase (a) 10 tickets or (b) 15 tickets?

For the lottery described in Exercise 2.4-13, find the smallest number of tickets that must be purchased so that the probability of winning at least one prize is greater than (a) 0.50; (b) 0.95.

A hospital obtains 40% of its flu vaccine from Company A, 50% from Company B, and 10% from Company C. From past experience, it is known that 3% of the vials from A are ineffective, 2% from B are ineffective, and 5% from C are ineffective. The hospital tests five vials from each shipment. If at least one of the five is ineffective, find the conditional probability of that shipments having come from C.

A company starts a fund of M dollars from which it pays $1000 to each employee who achieves high performance during the year. The probability of each employee achieving this goal is 0.10 and is independent of the probabilities of the other employees doing so. If there are n = 10 employees, how much should M equal so that the fund has a probability of at least 99% of covering those payments?

Your stockbroker is free to take your calls about 60% of the time; otherwise, he is talking to another client or is out of the office. You call him at five random times during a given month. (Assume independence.) (a) What is the probability that he will take every one of the five calls? (b) What is the probability that he will accept exactly three of your five calls? (c) What is the probability that he will accept at least one of the calls?

In group testing for a certain disease, a blood sample was taken from each of n individuals and part of each sample was placed in a common pool. The latter was then tested. If the result was negative, there was no more testing and all n individuals were declared negative with one test. If, however, the combined result was found positive, all individuals were tested, requiring n+1 tests. If p = 0.05 is the probability of a person’s having the disease and n = 5, compute the expected number of tests needed, assuming independence.

Define the pmf and give the values of \(\mu\), \(\sigma^2\), and \(\sigma\) when the moment-generating function of X is defined by (a) \(M(t) = 1/3 + (2/3)e^t\). (b) \(M(t) = (0.25 + 0.75e^t)^{12}\).

(i) Give the name of the distribution of X (if it has a name), (ii) find the values of and 2, and (iii) calculate P(1 X 2) when the moment-generating function of X is given by (a) M(t) = (0.3 + 0.7et )5. (b) M(t) = 0.3et 1 0.7et , t < ln(0.7). (c) M(t) = 0.45 + 0.55et . (d) M(t) = 0.3et + 0.4e2t + 0.2e3t + 0.1e4t . (e) M(t) = 10 x=1 (0.1)etx.