Discussion

Random variables are all around you. For example, a random variable could be the number of minutes that you spend on the phone each day or how many times you check your email each day. For this assignment, you will participate in a discussion about random variables.

Instructions

1. Complete the following on the Discussion Board:

a. Define the term random variable.

b. Post three possible random variables that you encounter in everyday life. An example could be the number of calories you consume each day. Some days you may consume more calories, while other days you may eat less and consume fewer calories.

c. Describe each of your examples and explain how they fit the characteristics of being a random variable.

2.1 Assignment

Many decisions in life are based upon uncertainty. However, if you know the probability, you may be able to make a more informed decision. For this assignment, you will answer questions and solve problems involving basic probabilities.

Instructions

1. Complete the following in a Word document:

· Write an original definition of probability based on what you have read.

· Write an original definition of sample space based on what you have read.

· Write an original definition of event based on what you have read.

· Write an original definition of probability distribution based on what you have read.

· Write out the sample space for a single toss of a fair coin.

· Write out the probability of rolling an odd number if you are rolling a regular six-sided die.

· Write out the probability of rolling a number greater than 4 if you are rolling a regular six-sided die.

· Write an event where the probability of that event is 0 if you are rolling a regular six-sided die.

2.2 Binomial Distributions

For this assignment, you will answer questions and solve problems involving binomial probabilities.

Instructions

1. Answer the following questions in a Word document:

· Is the binomial distribution a discrete probability distribution or a continuous probability distribution? Explain.

· If you are tossing a fair coin 10 times, what is the probability of getting exactly 4 heads out of the 10 coin tosses?

· If you are tossing a fair coin 10 times, what is the probability of getting exactly 9 heads out of the 10 coin tosses?

· If you are tossing a fair coin 10 times, what is the probability of getting 4 OR 5 heads out of the 10 coin tosses?

· The probability that an archer hits a target on a given shot is .7. If five shots are fired, find the probability that the archer hits the target on three shots out of the five.

· The probability that an archer hits a target on a given shot is .7. If five shots are fired, find the probability that the archer doesn’t ever hit the target during the five shots.

· The probability that an archer hits a target on a given shot is .7. If five shots are fired, find the probability that the archer hits the target on all five shots.

2.3 Probability and Distributions

Consider the experiment of tossing a fair coin four times. The coin has two possible outcomes, heads or tails.

a. List the sample space for the outcomes that could happen when tossing the coin four times. For example, if all four coin tosses produced heads, then the outcome would be HHHH.

b. If each outcome is equally likely, what is the probability that all four coin tosses result in heads? Notice that the complement of “all four heads” is “at least one tail.” Using this information, compute the probability that there will be at least one tail out of the four coin tosses.

1. Suppose you roll a single fair die and note the number rolled.

a. What is the sample space for a single roll of a fair die? Are the outcomes equally likely?

b. Assign probabilities to the outcomes in the sample space found in part (a). Do these probabilities add up to 1? Should they add up to 1? Why?

c. What is the probability of getting a number less than 4 on a single roll?

d. What is the probability of getting a 1 or a 2 on a single roll?

3. Suppose we are interested in studying movie ratings where movies get rated on a five star scale. One star means the critic thought the movie was horrible, and five stars means the critic thought it was one of the best movies of the year. Here is a frequency table for all the movies rated by this critic for the year:

a. Using this information, if we chose a movie from this group at random, what is the probability that the movie received a:

· 1 star rating?

· 2 star rating?

· 3 star rating?

· 4 star rating?

· 5 star rating?

b. Do the probabilities from part (a) add up to 1? Why should they? What is the sample space in this problem?

4. Given P(A) = 0.6 and P(B) = 0.3

a. If A and B are mutually exclusive events, compute P(A or B).

b. If P(A and B) = 0.2, compute P(A or B).

c. If A and B are independent events, compute P(A and B).

d. If P(B|A) = .1, compute P(A and B).

5. Consider the following events for a college professor selected at random:

A = the professor has high blood pressure

B = the professor is over 50 years old

Translate each of the following scenarios into symbols. For example, the probability a professor has high blood pressure would be P(A).

a. The probability a professor has low blood pressure.

b. The probability a professor has high blood pressure and is over 50 years old.

c. The probability a professor has high blood pressure or is over 50 years old.

d. The probability a 40-year-old professor has high blood pressure.

e. The probability a professor with high blood pressure is over 50 years old.

f. The probability a professor has low blood pressure and is over 50 years old.

Rating Number of movies that got that rating

1 Star 28

2 Star 123

3 Star 356

4 Star 289

5 Star 56

6. Suppose we did collect data by asking professors how old they were and measuring their blood pressure. The table below reflects the data collected based on these two variables:

Low Blood Pressure High Blood Pressure Total

50 and Under 64 51 115

Over 50 31 73 104

Total 95 124 219

Let us use the following notation for events: U = 50 and under, O = over 50, L = low blood pressure and H = high blood pressure.

a. Compute P(L), P(L|U) and P(L|O).

b. Are the events L = low blood pressure and U = 50 and under independent? Why or why not?

c. Compute P(L and U) and P(L and O).

d. Compute P(H) and P(H|U).

e. Are the events H = high blood pressure and O = over 50 independent? Why or why not?

f. Compute P(L or U).

7. Ryan is a record executive for a hip hop label in Atlanta, Georgia. He has a new album coming out soon, and wants to know the best way to promote it, so he is considering many variables that may have an effect. He is considering three different album covers that may be used, four different television commercials that may be used, and two different album posters that may be used. Determine the number of different combinations he needs in order to test each album cover, television commercial, and album poster.

8. Which of the following are continuous variables, and which are discrete?

a. Number of heads out of five coin tosses

b. Qualifying speed for the Daytona 500 in miles per hour

c. Number of books needed for a literature class

d. your weight when you wake up each morning

9. A number of books were reviewed for a history class based on the following scale from 1 to 5: 1=would not recommend the book, 2=cautious or very little recommendation, 3=little or no preference, 4=favorable/recommended book, 5=outstanding/significant contribution.

Book Rating, x P(x)

1 .051

2 .099

3 .093

4 .635

5 .122

Suppose a book is selected at random from this group.

a. Is this a valid probability distribution? Why?

b. Find the probability that the book received a rating of at least 2. How does this probability relate to the probability that book received a rating of 1?

c. Find the probability that the book received a rating higher than 3.

d. Find the probability that the book received a rating of 3 or higher.

e. Compute the average or expected rating of the books in this group.

f. Compute the standard deviation for the ratings of the books in this group.

10. Consider a binomial experiment with n = 8 trials where the probability of success on any single trial is p = 0.40.

a. Find P(r = 0).

b. Find P(r >= 1) using the complement rule.

c. Find the probability of getting five successes out of the eight trials.

d. Find the probability of getting at least four successes out of the eight trials.

11. Suppose ten people are randomly selected from a population where it is known that 22 percent of the population are smokers.

a. For this example, define what a trial would be, what a success would be, and what a failure would be. Also, state the values of n, p and q for this example.

b. What is the probability that all ten of the people are smokers?

c. What is the probability that none of the ten are smokers?

d. What is the probability that at least three of them are smokers?

e. What is the probability that no more than two of them are smokers?

f. What would be the average or expected amount of smokers out of a sample of ten people from this population?