Ch. 11. Resources

Chapter 11: Understanding Randomness

In the example that follows, I will focus on the main type of problem we will consider in Chapter 11: namely, one that uses random numbers to simulate a complicated real-life event. In later chapters we will learn more rigorous mathematical techniques to compute the likelihood of various outcomes for such situations—at which time we'll use the term "probability." (If you've already learned how to do some simple probability computations, pretend that you haven't for the duration of this chapter and instead use the random simulation techniques to answer the HW questions.)

We'll look at simulations because: a) setting up simulations lends us valuable insight into the underlying mathematical problem; b) simulation allows us to estimate the answer when an abstract mathematical calculation would be difficult or impossible; and c) simulation gives us practice working with random numbers, which are useful in a variety of circumstances.

Chapter Quizzes

Consider the Chapter Quizzes here on WAMAP. If you have attempted these quizzes more than once, you may have realized that are different (but similar) questions that are offered randomly to each student. In some cases, there are infinitely many possible problems; in other cases, only a handful. Let's consider a quiz problem with five possible questions that is set up to randomly select one of these five questions whenever a student takes the quiz.

But what does "random" really mean? As you might expect, in this case it could mean that any time a student attempted the quiz, that student would have an equal likelihood of getting any one of the five possible matching questions.

Suppose for a moment this were a regular face-to-face lecture class and the quiz was being given in traditional pencil-and-paper format. In this case if I still wanted to randomly select one of the five possible questions for each individual student, I might roll a standard six-sided die and give each student the question (1, 2, 3, 4 or 5) corresponding to the number showing on the top face of the die. Of course, this would work a lot better if there were six possible questions, but I could always state in advance that if I rolled a 6 I would roll again. As long as the die wasn't "loaded" we might expect that each face of the die had an equal chance of showing, hence each student would have an equal chance of getting any one of the five quiz questions.

Now suppose I forgot to bring a six-sided die to class. Fortunately I never forget my calculator, so we can use that instead. Press the MATH button:

Then move the cursor over so that PRB (for probability, since that's where were headed with all this, eventually) is highlighted, then move down to randInt(:

Now press ENTER. Next type 1,5) [a 1 then a comma then a 5 and then a right parenthesis] so that you see randInt(1,5) on the screen and press ENTER. The TI-84 should give you a random integer from 1 to 5:

Except that for most of you it probably gave you a 5. (This works a lot better in a lecture class when I call out the numbers from 1 to 5 and hardly anyone raises their hand until we get to 5; trust me, it makes an impression.) So, what's up? Well, the TI-84 doesn't give us truly random numbers, rather it gives us pseudorandom numbers.

To get around this do the following. Make up a 4- or 5-digit number. This is your seed. Mine will be 12345. (Don't choose this one!) Now type this number into the calculator, then press STO> (this is the "store" button, above the ON key):

then MATH, move over to PRB, and press ENTER. You should see 12345→rand (or your seed, an arrow and rand) on the screen.

Now press ENTER again:

This number has been stored as the random seed on your TI-84. Now you won't get the same answers as all of your classmates the next time we use the randInt feature.

Getting back to our example, suppose we wanted to simulate assigning one of five quiz problems to all 30 students enrolled in the class. We could use randInt(1,5) 30 times in a row. This isn't quite as bad as it seems, because after you enter it the first time you can keep pressing ENTER 29 more times until you have 30 numbers:

Or you could type randInt(1,5,30) to get 30 random numbers from 1 through 5, all at once:

But as you can see, you have to scroll to the right repeatedly to see all 30 numbers. So you might want to use randInt(1,5,6) and press ENTER 4 more times so that you get 6×5 = 30 numbers:

Remember, though, that the TI-84 doesn't give us truly random numbers. For most of the exercises we work in class that won't really be a problem, but it is for some people, for whom it is vital that the numbers they use be truly random. Where can we get truly random numbers? Where we get everything these days, the Internet!

True random numbers

Go to random.org and you will see a link that says Integer Generator:

Click on this link. We want to generate 30 random integers from 1 to 5 and we can format them in 6 columns:

Click Get Numbers and you should see something like this:

although your numbers shouldn't all be the same as mine!

Suppose now that on the quiz problem in question, no one was given question #4 by WAMAP. Should I be concerned about WAMAP's "random" quiz generator? I would need to know how likely it would be for the number 4 not to show up in this simulation that we just ran. So I could run the simulation on random.org another 199 times, say, and count the number of times 4 didn't show up.

But the real question I want to ask is, how likely is it that any one of the five numbers doesn't show up out of a group of 30. (After all, I would have been just as concerned if it had been #3 or #5 that was never assigned.) So instead of looking for just #4, I would look at these 200 simulations and count the number of times that at least one of the five numbers was missing from the group of 30 digits.

In fact, I did just that (with the help of a spreadsheet and some time-saving Excel formulas to test if one or more of the 5 digits was missing). Exactly 1 time out of 200 one of the digits was missing. So I will estimate that one of the quiz questions would not be assigned about 0.5% of the time. (In fact, the exact answer, which we'll learn how to compute later, is about 0.62%.) Might this raise my suspicions about WAMAP? Yes, since it seems like it should be a relatively rare occurrence, but at the same time I wouldn't conclude with certainty that there was a problem (since I would expect it to happen about 1 out of every 200 times I assigned the quiz problem to a class of 30 students).

This is just one type of simulation that we might wish to run. Another is covered in the example in Chapter 11, and many more arise in the Chapter 11 exercises. You can only truly get a feel for running a simulation by actually doing it, so you should work as may exercises as you can.

Homework

Work the following exercises in Chapter 11: 11, 17, 19, 25 and 39.

Errata

In the TI-83/84 Plus instructions on page 299, the last of the comments should read RandInt(0,56,3) (not 57) and the next line should read "0 to 56" (not "0 to 57").

In the TI-89 instructions, the first of the comments should read RandInt(0,1) (not 10) and the next line should read "a coin toss" (not "10 coin tosses").

Exercise 17 should read "Estimate how often you end up" (not "the probability that").

In the Just Checking answers on page 302, Exercise 2 should read "home team winning" and "visitors winning"' Exercise 5 should read "the number of wins" (not "proportion").

ActivStats

You may wish to work the activities on page 11-1 in the ActivStats lesson book, as time permits.

Additional Resources

random.org

True random numbers, served up fresh each day. When possible, use this site rather than your calculator (or Excel) to generate random numbers.

"Take a Chance: Scientists put randomness to work"

Erica Klarreich.
Science News. Vol. 166 No. 23. December 4, 2004. p. 362.
An article about how scientists employ randomness, including a discussion of random.org.

"Toss Out the Toss-Up: Bias in heads or tails"

Erica Klarreich.
Science News. Vol. 165 No. 9. February 28, 2004. p. 131.
An article about research conducted by Persi Diaconis at Stanford University to investigate whether flipping a coin is truly random.