STAT 280 Lab 2: The Dude's Probability Lesson

The final report for this lab is due one week from now.  Please write up your answers to the questions in a neat, organized fashion and turn in to your lab instructor.

Preliminaries

Excel data file is located at: http://www.stat.rice.edu/~hflores/stat280_gmdavis/, as stat280L5.xls

Goal for this lab

To understand how to perform probability calculations and (hopefully) make connections with statistical thought.

Setup 1 (population known, sample unknown)

So a dude walks up to you and hands you a bag 'o balls.  He sneers, "hey mac, if I were to draw a ball from this bag how likely do you think it would be for it to be any one color, eh?"  Disgusted with this fool, you grab the bag out of his hands, turn the bag over and dump its contents on the ground.  You see 30 multicolored balls.  Deciding that you had best make a list of the balls, you write down a number for each color of ball as follows (see data file):

1 = red
2 = yellow
3 = green
4 = blue
5 = black

The dude then proceeds to put all the balls back in the bag.

Answer the following questions:

1. If you were to reach into the bag and pull out 1 ball, what is the P[draw a red ball]?  P[draw a blue ball]?  (Drawing 2 balls, one after the other) P[draw a blue ball AND draw a red ball]?  (Drawing only 1 ball again) P[draw a red ball OR draw a yellow ball OR draw a black ball]?  

2. Define X as a random variable representing the event that you reach into the bag and pull out 1 ball (as in (1)).  What kind of random variable is this? (hint: there are only two types)  Why?

3. Give two ways to compute P[draw a blue ball OR draw a green ball OR draw a yellow ball OR draw a black ball]. HINT: Consider Rules 3 and 4, page 290 in your book.

4. You will now draw 5 balls one at a time, replacing each ball in the bag and mixing them up after each draw.  Compute P[ball1 = red AND ball2 = green AND ball3 = black AND ball4 = green AND ball5 = yellow].  Re-compute this probability when you DO NOT replace each ball in the bag after each draw.  Are these numbers different enough to convince you that drawing WITH and WITHOUT replacement is an important distinction?  Why?

Notes: It is important to understand what is going on in this example.  Realize that the dude is asking for you to give him information about a sample drawn from a ball population.  Recall that you dumped the bag's contents on the floor and collected everything you knew about the ball population, so calculating probabilities is easy when you know everything about the population.  This exactly the opposite problem in the world of statistics where we do not know everything about the population and have to make inferences based on a sample.  So, to summarize, when you know all about the population you can make inferences about samples drawn from it (what we just did in Setup 1).  However, when you don't have the population to "dump out on the floor and count" so to speak, you must take a sample that reflects the true population to help you make inferences about the population.  As such, Probability (the art of making inferences on samples based on a known population) and Statistics (the art of making inferences about populations based on known samples) are intimately related fields. 

Setup 2 (population unknown, sample known)

The same dude walks up to you with same bag o' balls.  He sneers, "hey mac, as it turns out I drew these balls randomly and independently from a huge crate of balls.  I wanna know more about what kind of balls are in the crate, but I don't wanna have to dump'em out and count 'em all.  So can you use this bag o' balls to tell me something about what's in the crate?"

We know think of our bag o' balls data in a different way, as a sample instead of a population.

Answer the following questions:

1. What important information did the dude give you to help you decide that his bag o' balls is a representative sample from the population?  Why is this important?

2. Compute a frequency histogram of the data.  Compute the median and mode.  What kind of ball would you say is the most prevalent in the population (crate of balls) if you had to guess?

3. Suppose I told you that the crate had 25000 balls in it.  Predict how many balls of each type there are in the crate based on the data.

4. Is it possible that there are balls of other types (not any of the colors collected) in the crate?  What would you tell the dude to do to be more certain?

Summary Questions (answer quasi-subjectively and creatively):

1. Since we are using the same data in both setups, what is really different between setup 1 and setup 2?
2. What other statistics (or plots) might have been useful in analyzing setup 2?