Flipping Coins and Exponential Decay
Lesson 2 of 7
Objective: SWBAT develop a tangible understanding of the distinction between experimental and theoretical values in probability.
Today's opener is just like yesterday's, except that now we're going to calculate the probabilities of pulling two cards from a deck. That's more complicated than drawing one card, and between today's opener and tomorrow's lesson, kids will get a sense of why.
Just like yesterday, students will find a handout at each table (U4 L2 Opener Handout), with six questions about drawing cards from a deck, and they will spend a few minutes in groups trying to come up with solutions to the problems. I'll give kids anywhere from 2 to 10 minutes to work on these problems. If engagement is high, I'll let everyone keep working. If I see that kids are stuck, I'll start a whole-class discussion sooner. Either way, each of these six problems is on a slide of the lesson notes.
Unlike yesterday, I don't have the solutions ready to go on these slides. I want to work through the solutions with kids. For the first two (Opener #1 of 3 and Opener #2 of 3), the sample space is small enough to sketch some quick tree diagrams. Most kids can remember seeing this before, and it's enough to jog their memories about why multiplication works: "if there are four possibilities for the suit I pick first, and for each of those possibilities, there are four possible suits to draw next, then that's sixteen different possibilities." On the third problem, and those that follow, I transition toward using more formal probability notation, and we get at the multiplication rule. All along we're also referencing what we did yesterday.
You can think of this and yesterday's openers as the start to tomorrow's lesson, in which we'll see why combinations and permutations exist. I find that it's important to spend a little time on this sort of compound probability with replacement - independent events - before jumping into drawing two cards at once - i.e. dependent events and permutations. Today's examples give students a chance to solidify the concept of multiplying simple probabilities to arrive at compound probabilities of independent events. We'll build on that when we start to consider what else changes when we talk about drawing two or more cards at the same time. I have to see what kids know and meet them there. If the idea of multiplying probabilities is completely foreign to kids, then I'll adapt to that. On the other hand, when many students show that they're familiar with problems like these, I know that it will be that much easier to get started on tomorrow's lesson.
Randomness and Expectations
As we continue to jump into this unit about probability, we have to talk a little bit about randomness and our expectations of what might happen in a given situation. In order to get kids thinking about these ideas, I show two videos from Veritasium. I guarantee that a brief perusal of their YouTube page will help you generate lesson ideas -- or at least give you a few cool anecdotes that you'll find yourself drawing on at the most unexpected times during your lessons.
The first video (What's My Rule?) is about expectations, and the idea that we must be careful to consider what we expect versus what might really be true. This video provides a simple example of how all of us are predisposed to confirm our expectations rather than challenge them, and that can get in the way of insight. During this unit, students will have to be able to put their expectations aside and look openly at what actually happens. Today we'll put words to these ideas when we talk about empirical and theoretical probabilities.
The second one is food for thought as we dig into the role randomness plays in probability. The video is not specifically about probability, but it gets kids thinking about the nature of randomness.
- What do we really mean when we say something is random?
- If massive amounts of information are not random, then what can we do about that?
- How much do do we need to see, in order to make an appropriate prediction?
I don't ask kids to do anything formal with these videos, just to watch and to think. The connections between these and the experiments of this and the previous lesson will come naturally. As we get into today's experiment, students will find themselves referencing what we've seen here -- randomness and expectations play an important role in the work we'll do.
Today, students are going run multiple trials of a coin-flipping experiment. Just like they did yesterday, students will run multiple trials of the experiment, collect data from each trial, and then aggregate those results. In contrast to yesterday's lesson, where we didn't know exactly what to expect, it's pretty easy to say what we'd expect today: if we flip 100 pennies, we expect approximately half of them to come up heads, and half to come up tails. I talk about some of these ideas and give an overview of the task in this narrative video. The key is that actually doing the experiment makes tangible the idea of expected, or theoretical, results versus experimental, or empirical, results.
As I've described in the video, the experiment is pretty straightforward:
Each group of students will receive a cup filled with 100 pennies. They will shake the cup and dump all 100 pennies on a table, then separate the coins by whether they came up heads or tails, keeping the heads, and setting aside the tails. Students record the number of heads after each "flip", then they put those back in the cup, and repeat until there are no coins left.
Each group will run three trials of today's experiment, and record their data on this handout. Next, they will compute the average number of coins remaining after each flip for those three trials. Then, on the back of the handout, students will collect the average results from each other group, and average these averages to calculate results for our whole class. This part of the lesson requires students to move around the room and talk to each other, and I love listening to - and occasionally jumping in on - the conversations that spring up.
If there are six groups and each runs three trials, then the whole class averages will be based on the results of 18 trials. At this point, I can ask, "As a whole class, do our results approach expectations?" Here we have to think about what we'd expected in the first place. As a group, we develop the understanding that we expect half of the coins to "be eliminated" every time we flip them. Finally, we can compare these out results to our expected values.
What's really interesting is that although it's impossible to have a fraction of a coin after an individual step, the aggregate results of many trials start to approach the theoretical value of 12.5 after three flips. Compare this to the statistic that the average American family has 2.2 children. Does any individual family have 2.2 kids? No - we can see the problem with that!
I include the results of one class running the experiment 20 times. I colored the values to show when each trial is complete. The colors help to produce an informal data visualization that we'll be able to reference when we start construction probability distributions a few days from now.
After collecting the data, I ask my students to create graphs of their data. I distribute graph paper and instruct all students to graph each trial in a different color. To help with the visualization, we might choose to connect the dots. Even though this is discrete data, connecting the dots can help us see more clearly the shape of each curve (see graphs without lines and graphs with lines).
Now, students should recognize the shapes of these graphs from Unit 3 - this is an instance of exponential decay. (I should note that I've considered including this lesson in Unit 3, but the ideas of probability and sampling are big enough that I run the lesson here, and I consider the use of exponential modeling to be a review.)
After students graph their three trials, I ask everyone to try to write a function rule (or a recursive definition, if they're so inclined) that represents our expectations in this problem, and then to graph that as well. We see that all trials follow that model approximately, with only minor or moderate deviation. Finally, because we're looking deeply at the importance of running multiple trials, students can graph the average results as well. In all likelihood, those average results will be tough to distinguish from the expected results, especially at the level of precision afforded by our graph paper.
To debrief, we think big: what if we could run this experiment 100 times or 1000 times?You can check out the Simulation via Java Program reflection that follows to see how I introduced a computer simulation of this same experiment to extend the lesson when time allowed for an extension.
This Exit Slip helps me assess what students understand after the coin flipping experiment. I've tried to squeeze a lot in today; if I have to save this task for tomorrow, that's fine. I'm just including it here to show how I assess student understandings of today's work. There's nothing special in my delivery here - I just provide the handout, and give kids 10 minutes or so to work on it.
To answer the 1st and 4th questions, I expect students to show me what they've learned about theoretical probability today, and to apply it to a similar situation with slightly different parameters. I don't tell kids to use theoretical probability, because part of what I'm looking for is whether they know when to use theoretical probability as a tool for answer certain questions.
The second and third questions require students to apply what they've seen about how real life will often break our expectations on individual occasions. Students might have a chance to mention something they saw in one of the videos earlier in today's class here. Whatever they tell me, I'm looking for whether or not they can grasp what is possible in the context of a given situation.
I like this prompt because it's easy to change it. We can talk about saving more or fewer dice after each roll, and how that might change the outcome. If we need to review the parameters of exponential decay functions, here's a chance to that: how would the graph change if we only kept the 5's each time? Everything but the 5's? Even numbers? How can we make this experiment most similar to the coin flipping experiment?