1) A door-to-door salesman expects to make a sale 26% of the time when starting the day. But making a sale increases his enthusiasm so much that the probability of a sale to the next customer is 0.36. If he makes no sale, the probability for a sale to the next customer stays at 0.26. What is the probability that he will make at least two sales with his first three visits?

2)Two machines turn out all the products in a factory, with the first machine producing 50% of the product and the second 50%. The first machine produces defective products 2% of the time and the second machine 7% of the time.

(a) What is the probability that a defective part is produced at this factory given that it was made on the first machine?


(b) What is the probability that a defective part is produced at this factory?

3)Dystopia county has three bridges. In the next year, the Elder bridge has a 11% chance of collapse, the Younger bridge has a 2% chance of collapse, and the Ancient bridge has a 22% chance of collapse. What is the probability that exactly one of these bridges will collapse in the next year? (Round your final answer to four decimal places. Do not round intermediate calculations.)

4. A subway has good service 70% of the time and runs less frequently 30% of the time because of signal problems. When there are signal problems, the amount of time in minutes that you have to wait at the platform is described by the pdf probability density function with signal problems = p_T|SP(t) = .1e −.1t.

But when there is good service, the amount of time you have to wait at the platform is probability density function with good service = p_T|Good(t) = .3e −.3t.

You arrive at the subway platform and you do not know if the train has signal problems or running with good service, so there is a 30% chance the train is having signal problems.

(a) After 1 minute of waiting on the platform, you decide to re-calculate the probability that the train is having signal problems based on the fact that your wait will be at least 1 minute long. What is that new probability?

(b) After 5 minutes of waiting, still no train. You re-calculate again. What is the new probability?

(c) After 10 minutes of waiting, still no train. You re-calculate again. What is the new probability?

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Do you take the free samples offered in supermarkets? About 62% of all customers will take free samples. Furthermore, of those who take the free samples, about 37% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 317 customers passed by your counter. (Round your answers to four decimal places.)

(a) What is the probability that more than 180 will take your free sample?


(b) What is the probability that fewer than 200 will take your free sample?


(c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.37, while P(sample) = 0.62.


(d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Do you take the free samples offered in supermarkets? About 56% of all customers will take free samples. Furthermore, of those who take the free samples, about 37% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 329 customers passed by your counter. (Round your answers to four decimal places.)

(a) What is the probability that more than 180 will take your free sample?

(b) What is the probability that fewer than 200 will take your free sample?

(c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.37, while P(sample) = 0.56.

d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. Do you take the free samples offered in supermarkets? About 58% of all customers will take free samples. Furthermore, of those who take the free samples, about 35% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 323 customers passed by your counter. (Round your answers to four decimal places.) (a) What is the probability that more than 180 will take your free sample? (b) What is the probability that fewer than 200 will take your free sample? (c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.35, while P(sample) = 0.58. (d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

A subway has good service 70% of the time and runs less frequently 30% of the time because of signal problems. When there are signal problems, the amount of time in minutes that you have to wait at the platform is described by the pdf probability density function with signal problems = pT|SP(t) = .1e^(−.1t). But when there is good service, the amount of time you have to wait at the platform is probability density function with good service = pT|Good(t) = .3e^(−.3t) You arrive at the subway platform and you do not know if the train has signal problems or running with good service, so there is a 30% chance the train is having signal problems. (a) After 1 minute of waiting on the platform, you decide to re-calculate the probability that the train is having signal problems based on the fact that your wait will be at least 1 minute long. What is that new probability? (b) After 5 minutes of waiting, still no train. You re-calculate again. What is the new probability? (c) After 10 minutes of waiting, still no train. You re-calculate again. What is the new probability?

A company is considering submitting a tender for a job.
Producing the tender will cost the company $1500, which must be paid even if the company does not win the tender
The company's current expectation of the cost of the project is that there is a 0.58 probability that the cost is $7700, and otherwise the cost is $10900. These costs do not include the cost
of producing the tender.
The company also believes that their is a 0.28 probability that the lowest competing bid will be $9200, a 0.37 probability that the lowest competing bid will be $14400, and otherwise the
lowest competing bid will be $12000
In regard to the tender price the company is considering whether it should bid $10600, bid $13200, or not bid.
You may assume that the probabilities given for the costs of the project are independent to the probabilities of the values of the lowest competing bid. Thus if they submit a price of $13200
then the probability that they will make the maximum profit ($13200 - $7700 - $1500 = $4000) is P(winning) * P(low cost) = 0.37 * 0.58.
Draw a decision tree that represents this situation and show all EMVs and probabilities. Indicate selection of decision options with probability = 1 for preferred
options and probability = 0 for rejected options.
Monetary values should be answered to the nearest dollar. Probabilities should be answered to three decimal places.

1. Suppose it has been estimated that 60% of American victims of health care fraud are senior citizens and that 10 victims are randomly selected.

a) What is the probability that exactly 4 of the victims are senior citizens?

b) What is the probability that at least 8 of the victims are senior citizens?

c) Determine the expected value.

d) Determine the standard deviation.

2. Suppose that 9% of all men cannot distinguish between the colors red and green. This is the type of color blindness that causes problems with traffic signals. Suppose that 8 men are randomly selected.

a) What is the probability that exactly 2 of the men will have this type of color blindness?

b) What is the probability that at least one of the men will have this type of color blindness?

c) Determine the expected value.

d) Determine the standard deviation.

3. Suppose it has been estimated that 68% of all Albanian Airlines flights are on time, and that 12 flights are randomly selected.

a) What is the probability that fewer than three of the flights were on time.

b) What is the probability that more than ten of the flights were on time.

c) Determine the expected value.

d) Determine the standard deviation.

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. Do you take the free samples offered in supermarkets? About 58% of all customers will take free samples. Furthermore, of those who take the free samples, about 33% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 315 customers passed by your counter. (Round your answers to four decimal places.) (a) What is the probability that more than 180 will take your free sample? (b) What is the probability that fewer than 200 will take your free sample? (c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.33, while P(sample) = 0.58. (d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

One study has reported that the sensitivity of the mammogram as a screening test for detecting breast cancer is 0.88, while the specificity is 0.75

1. What is the probability of a false negative test result? Which answer is correct? There might be slight rounding differences. Please provide an explanation.

a. 50%

b. 100%

c. 23%

d. 12%

2. What is the probability of a false positive test result? Which answer is correct? There might be slight rounding differences. Please provide an explanation.

a. 25%

b. 75%

c. 100%

d. 0%

3. In a population in which the probability that a woman has breast cancer is 0.003, what is the probability that she has cancer given that her mammogram is positive? Which answer is correct? There might be slight rounding differences. Please provide an explanation.

a. 0.023

b. 0.534

c. 0.0011

d. 0.011

4. In a population in which the probability that a woman has breast cancer is 0.0015, what is the probability that she has cancer given that her mammogram is positive? Which answer is correct? There might be slight rounding differences. Please provide an explanation.

a. 0.1234

b. 0.4321

c. 0.6754

d. 0.0053