Mathematics

homework

1. One out of every eight calls to your house is from a family member. You will record whether the next call is from a family member.

(a)What do you consider a “success” in this story? What is its probability?

(b) Dene in words what X is in terms of this story. What values can X take?

(c) What is the distribution and parameters of X?

(d)What is the probability the next time the phone rings, it will be from a family member?

(e)If the phone calls are independent, what is the probability the 3rd call today will be from a family member?

(f) What is the mean of X?

(g) What is the standard deviation of X?

2. On a certain highway, 7% of the vehicles have 18 wheels, and the other 93% of the vehicles have 4 wheels. (We ignore motorcycles, etc., for simplicity.) A child looks out the window and counts the wheels on the next vehicle to pass.

(a) What is the expected number of wheels?

(b) What is the variance of the number of wheels?

3. Skittles candies come in the colors red, orange, yellow, green and purple, with each color having equal probability. You are a quality control inspector, and your job is to count up the number of purple candies in a random sample of 25 candies coming down the factory line.

(a) What is a “success” in this situation and what is the probability of a success on a single trial?

(b)Explain in words what X is in terms of the story and what values it can take on.

(c)What are the distribution and parameters of X?

(d) What is the probability you find exactly 5 purple candies in your sample?

(e)What is the probability that you will find at least 2 purple candies in your sample?

(f)What is the expected number of purple candies in the sample?

(g)What is the standard deviation in number of purple candies in the sample?

(h)If purple candies cost the company an extra 1.5 cents each to make with an additional general production fee of 25 cents (not related to the number of purple candies), what is the expected extra cost of purple candies for this sample?

(i) What is the standard deviation of the extra cost of purple candies in this sample?

4. Twenty percent of babies in a particular city are born by a surgical procedure called a

Cesarean section (C-section). You survey 9 parents of babies from the population of a large city and count the number of babies who are not born by C-section.

(a) Explain in words what X is in this situation and what values it can take?

(b) What are the distribution and parameters of X?

(c) What is the probability exactly 6 babies were not born by C-section?

(d) What is the probability that between 4 and 6 babies (inclusive) were not born by C-section?

(e) What is the probability that exactly 6 were not born by C-section if you knew between

4 and 6 babies (inclusive) were not born by C-section?

(f) How many babies would you have to survey in order to be at least 90% sure you would find at least 1 who is not born by C-section?

(g) Calculate the expected value and variance of X.

5. (Ward and Gundlach) Jeff typically makes 80% of his field goals. Steve typically makes 60% of his field goals. Suppose they both have the opportunity to kick 3 field goals.

(a)What is the probability Jeff will succeed in making at least 1 field goal?

(b)What is the probability Steve will succeed in making at least 1 field goal?

(c) What is the probability that both Steve and Jeff will succeed in making at least 1 field goal?

(d)What is the probability that either Steve or Jeff will succeed in making at least 1 field goal?

(e)Given that only one field goal total was scored, what is the probability that Jeff was the one who kicked it?

6. According to the Guinness Book of World Records (2005), there are 276 marriage licenses issued in Las Vegas per day. Let X be the number of marriage licenses issued in the next hour.

(a)What are the distribution and parameters of X?

(b)What is the expected number of marriage licenses issued in the next hour?

(c)What is the standard deviation of the number of marriage licenses issued in the next hour?

(d)What is the probability that between 10 and 12 (inclusive) marriage licenses will be issued in the next hour?

(e)What is the probability that between 10 and 12 (inclusive) marriage licenses will be issued in 12 out of the next 24 hours? State the distribution and parameters you are using.

7. According to the Guinness Book of World Records (2005), the fastest pumpkin-carver on record, Steven Clarke, carved 42 pumpkins an hour. Assume this is his average rate. Let C be the number of pumpkins Steven carves in an hour.

(a) What are the distribution and parameter of C?

(b) What is the probability Steven will carve exactly 40 pumpkins in the next hour?

(c) Given that he has carved at least 3 pumpkins in a 5-minute interval, what is the probability that he will carve at least 4 pumpkins during that 5-minute interval?

(d) Given that he carved fewer than 4 pumpkins in a 5-minute interval, what is the probability he carved fewer than pumpkins during that 5-minute interval?