1. You have an unfair coin that is designed to come up heads 80% of the time.

a.) What is the probability that you will have to flip the coin exactly three times to get the first head?

b.) On average, how many times must you flip it to get the first head?

c.) On average, how many times must you flip it to get the first tail?

d.) What is the probability that you must flip the coin four or more times to get the first tail?

2. You have a small company that makes fiberglass surfboards. The molding process is very tricky, and results in defects exactly 25% of the time, and this probability is independent of how many other defects there have been that week. You attempt to mold exactly 10 surfboards every week. Let the random variable X indicate the number of non-defective surfboards you produce in any given week.

a.) What are the possible values for X?

b.) Give the entire distribution (probability mass function) for X.

c.) What is the average number of non-defective surfboards you make per week?

d.) What is the variance in the number of surfboards you make per week?

e.) State completely the cumulative distribution function for X.

f.) What is the probability that you produce 7 or more non-defective surfboards in a given week?

g.) If the profit you make is $100 per non-defective surfboard produced in the week minus $450 in fixed costs, what is the average profit you make per week, and what is the variance?

3. You place 15 red poker chips and 10 blue poker chips in a container and shake them up. You ask your friend to take 18 chips from the container without looking. Let the random variable X represent the number of red chips he has taken.

a.) What is the set of possible values for X?

b.) State the entire probability distribution (pmf) for X.

c.) State the cumulative distribution function for X.

d.) What is the probability your friend has drawn 12 or fewer red chips?

e.) What are average and variance for the number of red chips your friend draws?

4. Returning to the unfair coin from question 1 (that comes up heads 80% of the time). Now, however, you are playing a game in which you keep flipping the coin until it comes up heads five times.

a.) What is the probability that you must flip the coin exactly 7 times in order to get 5 heads?

b.) What is the probability you will need to flip the coin 8 or more times to get 5 heads?

c.) What is the average number of times you will need to flip the coin to get 5 heads?

5. Suppose that the number of devastating earthquakes (above a certain magnitude and occurring in a populated region and/or producing dangerous tsunami) occurring somewhere in the world averages out to 2.5 per year. Furthermore, assume that the occurrence of devastating earthquakes is a Poisson process.

a.) What is the probability of having exactly 3 devastating earthquakes in any one year?

b.) What is the probability of having 1 or fewer such earthquakes in any one year?

c.) What is the probability of having 4 or more devastating earthquakes in a given year?

d.) What is the variance in the number of devastating earthquakes per year?

e.) Over any given 10-year period, what is the probability that there will be 2 or more years in which 4 or more earthquakes happen in each of those years?

f.) Over any given 10-year period, what is the probability that there will be 3 or more years in which 1 or fewer earthquakes happen in each of those years?

(Hint: for parts e and f, you will need to combine what you know about two different families of discrete distributions.)

6. You breed miniature mules (father is a miniature donkey, mother is a miniature horse) and currently have as breeding stock 1 jack (male donkey) and 7 mares (female horses) and these are able to produce 7 mule foals (baby mules) for you every year. You would like to produce miniature mules that are quite small, but unfortunately there is considerable variation in the sizes of mule offspring produced even from the same set of parents. Based on the best information you can obtain, the heights of the mules you produce are normally distributed with a mean height of 32.5 inches and a standard deviation of 1.25 inches. Any mule less than or equal to 32 inches can be sold for a high price and any mule greater than 32 inches in height can be sold only for a lower price.

a.) Assuming that your model (normal distribution with parameters as stated above) of mule heights is accurate, and also assuming that there is no limit to the level of precision with which you can measure the mules, what is the probability that any given mule will turn out to be one you can sell at the high price?

b.) What is the probability that any given mule born on your farm will fall between 31.25 and 33.75 inches in height?

c.) What is the probability that any given mule of yours will fall between 28.75 and 36.25 inches?

d.) What is the median height of your mules? What is the 40^th percentile height of your mules?

e.) What is the probability that out of the 7 mules born on your farm in any given year, at least 3 of them will be of the size that can be sold for the high price?

f.) What is the average number and standard deviation in the number of mules per year that can be sold for the high price?

g.) Of the 7 mules born in any given year, what is the probability that at least 4 of them will fall between 31.25 and 33.75 inches?

h.) Of the 14 mules born in any two year period, what is the probability that at least 1 will be less than 29.5 inches?

i.) If a mule foal is both less than 32 inches in height and of a desirable color it can be sold for a very high price. You have determined that there is a 20% probability that any given foal will be of a desirable color, and that color is independent of height. What is the probability in any given year that you will have at least one mule that can be sold for a very high price in any given year?

7. Assume that the fish of a particular species in a particular river range from 10 to 15 inches in length, and that the distribution of their sizes is uniform. Youre not a catch-and-release fisherman, and you really do want to take home six of these fish for supper for your family. However, to keep the fish, it must be at least 12 inches in length, and you must release any fish smaller than this. Assume that the number of fish in the river is very large, so you need not be concerned about issues of drawing without replacement.

a.) Express the probability density function for the distribution of fish sizes.

b.) Express the cumulative distribution function for the fish sizes.

c.) What is the average size and the standard deviation in the sizes of the fish? What is the median size?

d.) On average, how many fish do you expect you will have to catch before you have six that you could keep? What is the standard deviation?

e.) What is the probability that youll have to catch 10 or more fish before you will have six that are big enough to take home?

8. Your friend also likes to fish, but he is more impatient, and he does not really need to take a certain number of fish home. He catches on average 2.5 fish per hour, and this seems to be a Poisson process.

a.) What is the average amount of time in minutes he must wait between catching one fish and the next?

b.) What is the standard deviation of the time he must wait between catching one fish and the next?

c.) What is the cumulative distribution function of his waiting times between catching one fish and the next?

d.) Hell always catch his first fish of the day, but after that, since he is impatient, he will quit and go home if he ever has to wait more than 30 minutes between catching fish. What is the probability that he will go home before he catches his third fish of the day (of any size)?