32 Probability and Binomial Simulator

Probability and Statistics intersect when you do inference. And some of the most important situations in probabiity are binomial. We have a plugin for that.

The Binomial Simulator lets you set up binomial situations and then choose a number of trials. The plugin emits the data from the random trials into a CODAP table where you can then analyze the data.

The especially cool thing about this plugin is that it pays attention to language. This is because students often have trouble disentangling the various parts of a binomial problem, and I (Tim) think that part of that is the language we use to describe the context.

What does that mean? I’ll give you a detailed example, and then a problem set you can use with students—or yourself.

32.1 The Aunt Belinda Problem

For me, Aunt Belinda is one of those prototypical problems that you can refer back to time and again. As you will see, it’s really also an inference problem; it’s structured just like a significance test, but it’s designed to be understandable as a first example of the genre. Here we go!

Your Aunt Belinda claims to have magical powers. She says she can make nickels come up heads when she flips them. You ask for a demo, and give her 20 nickels. She mumbles some magical-sounding words over the nickels and flings them into the air. You scurry around to pick them up and find that of the 20 nickels, 16 came up heads.

Is it magic?

We’re not sure, but let’s make the question a little different:

If Aunt Belinda had no powers, how often would we get 16 heads in 20 flips?

Of course, in the classroom, you do this with real coins and collect data from the students. You get maybe fifty trials and see that 16 heads is at least rare. But of course you want more data, so it becomes a job for a computer.

In the live illustration or separate link below, we have set up the simulator correctly for this problem. Notice:

Under Settings you have all the numbers for our investigation:
- The probability of heads is 1/2. You can put fractions, decimals, or percentages in this box.
- The number of trials, “20,” is listed under Settings as “flips per experiment.” That’s the number of coins Aunt Belina flips.
- The number of experiments—200 in this case—is arbitrary but should be large.
Under Vocabulary is the cool stuff: the words we would use to decsribe the different parts of the context. In this case, they’re all pretty clear except for maybe “experiment,” which could just as easily be “trial” or “set” or “run.”

Try this:

Click Engage to do the 20-flip experiment 200 times. A table appears.
Graph heads to see how many heads came up during each set of 20 tosses.
Observe how many times out of 200 you got 16 (or more) heads. Not very many!

Then, with the class, you have to ask the payoff question:

What does this result tell us about Aunt Belinda?

Accept all answers. Maybe she cheated. Maybe the coins were weighted. Maybe she really has magical powers. But guide the class to the payoff answer: either she was lucky, or something other than chance was working here.

If you want to go further,

Try to find the probability that you get 16 or more heads in 20 tosses of a fair coin. Here, I’m not advocating going into combinatorics, but rather finding a better empirical probabilty. To do that, you need more cases; simply increase that 200 number and/or press Engage more times. This could lead to a summarizing data move, creating a formula in some column that gets moved to the left.
Help students clarify why we’re interested in the probabilty of 16 or more in this Aunt Belinda context, instead of just the probability of 16.

32.1.1 Another way to make CODAP calculate for you

Don’t forget that CODAP has all kinds of useful tools in the graph. If you aren’t ready to make a summary column and drag it left, just do this:

Go to the eyeball palette and check Show measures for selection.
In the graph, go to the “ruler” palette and set it to report Count and Percent.

Then, when you select the extreme points, you can see how many there are and what percent they represent.

32.2 Exercises!

You have about a one in five chance of making a great artwork in a day. After five days, what is the chance that you have produced at least one great work?

Start in the vocabulary

The basic event in this case is a day. The results are great art or not so great. Then it says, what do you call a set of 10 days? Well, in this problem, it’s 5, not 10, right? Call it a week and change the 10 to 5. Now the description text says you’ll be making 100 weeks. That’s fine; increase it if you want. This is a fantasy: if I were to do 100 weeks, how much great art would I produce during each of those weeks? That’s what you’ll simulate in one run. Try it. Don’t forget to set the probability!)

Your plants grow 1 cm on sunny days and not at all on cloudy days. 5/6 of all days are sunny. How tall will the plants be after 30 days? What is a reasonable range for your prediction? (Again, the basic event is a day.)
Aloysius is taking a 10-item true-false test about which he knows nothing. What’s the chance that he will get a grade of 70%? (Or higher) (Basic event, a question.)
How much worse are his chances if it is a 20-item test?
100 people each play 50 games of roulette, betting one chip on red every time. How many of them are winners? What is the most that a player has won? How much did the house win altogether from these people? (you might call 50 games a session)
Suppose Grunt and Flaky, two Senate candidates, have equal numbers of supporters. You do a poll of 100 voters.
- What’s the chance that the poll shows one of them having 54% or more of the voters?
- Redo the problem with 1000 voters in each poll. Now what’s the chance of 54% or more?
Every day you have a 1% chance of sleeping through your alarm. What’s the chance that you go through a week without that happening? How about a month? An entire year?
Invent an entirely new binomial problem and solve it!

32.3 Teacher notes on the exercises

These problems should give you plenty of ideas for additional problems you—or your students—could write.

They also give you and your students ample opportunity to address other important issues that come up in stats and probability settings. Here are some:

Binomial problems let you discuss variability. Although there may be a clear expected number of “successes,” every trial can be different. For example, we expect the plants to grow 25 cm.

But do they? Sometimes they do, sure, but they usually grow some other amount.

Then there are issues around sample size. We’ve already mentioned how you need more trials to get a good value for the probability of 16 or more heads. But there’s another issue. Looking at Aloysius and his 20-item test, we see the chance of getting 70% or more goes down in comparison to his 10-item test. That’s because there’s less variability in the results (expressed as a percentage) from larger samples: his grade from random guessing will tend to be closer to 50% on 20 items than 10.

This exposes a point of confusion: in Belinda, we increased the number of experiments (the third Settings box) to get a more precise probability and increase the sample size. In Aloysius, we increased the number of questions in the test, and that number is in a different box.

So, we ask students, what is the difference between the numbers in those two boxes? Ultimately, the answer is (mostly) that the last box is all about precision, getting enough data points—and every “experiment” or “test” is a data point— to know the probability of some condition (heads ≥ 16, score ≥ 70%) more precisely.

The second box, on the other hand, is part of the context itself.

Empirical vs theoretical

I like doing these problems empirically, at least at first. The theoretical approach, with combinations and permutations and factorials, needlessly snows the students. This approach is more humane and understandable. But it is not without cost, and you pay it right here: in the theoretical approach, you never have to decide how many times to try it.

Roulette

Having said that, what about the Roulette problem? First, of course, someone has to learn about Roulette and figure out that (for standard US Roulette tables anyway) the probability of winning when you bet on red is 18/38, and that the payout is 1:1.

But then there’s the issue of what the “100” is doing in this problem. Here it naturally fits in the third Settings box, despite our saying that the number there is arbitrary, just good to be large.

Sure. Except the problem asks how many people out of 100 make money. So if you run the simulation once you get one answer. If you run it again, your answer may be different. So you could run it many times, and learn the distribution of the number of winners, the maximum winnings, and the house’s take.

That is, you can take the simulation another level up. It’s not just a set of 50 spins of the wheel, done 100 times, but rather {50 spins done 100 times} — done many times.

Grunt and Flaky: the Bayesian thing

Note that you have to describe the Grunt and Flaky problem carefully. The question we really want to ask is:

Suppose, in our poll of 100 voters, Grunt gets 54%. What’s the probabilty that he is actually ahead?

Notice that our question, “Suppose they’re tied, what’s the chance that Grunt appears to get 54%?” is a different question, and not really what we want to know.

The thing we really want to know is a Bayesian problem, not part of the traditional intro stats course.

32.4 Exploring the effect of sample size

Perhaps after you’ve done other things for a while, return to this binomial simulator and revisit the Aloysius problems above.

The general “Aloysius” question is something like this:

Aloysius is taking a true-false test with \(n\) questions. He actually knows nothing about the subject. What’s the chance that he passes the test with a score of 70% or better?

We’re looking for a function, really, \(P(n)\). But that’s too complicated here, so we use the simulator to find some empirical values for \(P\) for various values of \(n\). We will discover that the probability gets quite a bit smaller as \(n\) increases.

Notice, though, that you can find an exact answer for \(P\) for very small values such as \(n=1\) (0.5) and \(n=2\) (0.25).

A more general look at sample size

We may not care, ultimately, whether our students eventually solve these problems theoretically, using combinatorics. But I hope we care whether students develop a little intuition about sample size.

In particular, I think we want them to see how increasing sample size makes a result more precise. What do we mean by that? It’s actually kind of subtle.

Suppose we recreate Aunt Belinda, but use differing numbers of coins.

When we simulate 400 sets of 20 coins, we can make a graph with 400 points representing the number of heads. It’s a mound-shaped distribution, centered roughly on 10 heads.

Now do the same with 100 coins. You’ll get a very similar graph, but centered at 50 heads.

Now suppose we want to compare the two graphs more directly. We can make a single graph showing the two distributions, but it’s messed up because one graph goes from 0 to 20, while the other goes from 0 to 100.

We can make another column to plot: instead of heads, let’s calculate percentHeads.

Now look at the spreads in those distributions in the two graphs.

In the first graph, the small sample has the smaller spread.
In the second graph, the small sample has the larger spread.

You can even stick standard deviations on the graphs, or make box plots,
(use the ruler palette) and get precise numbers.

The takeaway should be: if you flip more coins, just by randomness, the number of coins will generally be further from 50%. But the percentage of coins should be closer.

That first observation goes with the “random walk” context: the longer you walk, on average, the farther you are from your starting point.

The second is an example of the law of large numbers: the more events you collect, the closer your estimate of the probability will be to the true probability.

Beyond coins

That’s all fine and abstract. But we want our students to be able to connect this understanding to real-world situations.

Eventually…

We should refer to a paper by Karin Binder and me about inference, but it’s not done yet!

Until then, a few words…

A great example is polling.

You can extend the Grunt and Flaky context up above, but change it like this:

Suppose, in a Senate contest bewteen Grunt and Flaky, Grunt actually, truly has the support of 52% of the voters.

You do a poll of 25 voters. What’s the probability that the poll shows Grunt winning? What’s the probability that the poll has Flaky with more than 55%?
Answer the same questions for a sample size of 100.
Now do a sample size of 400. Now a sample size of 1600!

Example results with Grunt (52%) and Flaky

Say in words what those percentages on the right really mean in terms of the context. Put another way: suppose you’re working for the Grunt campaign. What does this informtion tell you about the quality of infomation you can get from a poll?
Now record the standard deviations for the spreads in percentage for each sample size. Plot them against sample size—and see if you can find a function that predicts the spread from sample size.