Modeling Probability

This is a lecture class, but I put all the instructions in the handout as well, mostly so I can remember the outlines of the activity.

In the lecture, I explain that video games run on probabilities, that balancing probabilities is an essential element for strategy video games. Any games that give the user a running series of choices has to balance outcomes and create choices with tradeoffs. Otherwise, the user learns that the “good” choices are and the game gets boring. (I have no idea if this is true, but it sounds reasonable.)

Obviously, if the students were really at a software company, these scenarios would be automated, but hey, it’s a math class.

Materials: Scintilla Handout (reproduced here), and 9 Oracle advice cards (3 gryphons, 3 dragons, 3 excaliburs)

Here’s the main handout.

Crossing the Scintilla

You’re an intern at a major software development company! You’ve been assigned to work with the team developing Sorcery & Shadows, a new game scheduled for fall release.

S&S is “retro”, harking back to the 70s and 80s fantasy games—less violence, more strategy. At various points, the player’s choice of avatar must consult the three Oracles for permission and guidance. The Oracles determine the character’s actions, but the response varies based on the avatar chosen:

The software team is working on the following scenario:

The player (Vlad, Dulcinea, or Chaos) is trying to cross the Scintilla River. The Oracles must be consulted. The Oracles will each advise one of the following options:

1. The gryphon, which swims across.
2. The dragon, which flies across.
3. Excalibur, the sword, which magically transports the carrier.

Gryphon

Dragon                                         Excalibur

The player will follow the Oracles’ advice if the right number of them agree. The Oracles are pleased when the player follows their advice and give the player five silver coins.

If the player can’t follow the Oracles’ advice, then the player must pay 1 silver coin to cross on the ferry.

The software team has already decided that the Oracles’ responses are randomly generated, and they need to determine the probability that each character gets across the river. They’ve asked you to work this out.

Before you start, have a brief group discussion and make your predictions.

Is it equally probable that Vlad, Dulcinea, and Chao will be able to follow the Oracles’ advice?
1. If not, who do you think is the most likely to be able to follow the advice?
2. Least likely?

I wander around to listen in on the predictions. The two most common predictions are Vlad and Dulcinea, which is interesting.

I always do two trials as a class before I set them on their own, stressing that the actual advice—Gryphon, Dragon, Excalibur—is immaterial. They are tracking which character is able to accept the Oracle’s advice. Back to the handout—this is on the flip side.

Experimental Probability
Experimental Probability—Performing an event repeatedly and measuring the results as a ratio of a particular occurrence to the total number of trials: .

Once a pattern has been established, we can rely on this data as an empirical probability.

You are going to perform 30 simulated trials of the event “Asking the Oracles for Advice”.

Each group has three sets of three cards: a gryphon, a dragon, and Excalibur. Three group members will “play Oracle”, by randomly and simultaneously throwing down one of the three cards. (IT MUST BE RANDOM!). The other group member tracks the outcome of each trial, using the table below.

When you have completed 30 trials, compute the experimental probability of each character’s likelihood of following the Oracles’ advice. Remember to keep careful track of how many trials you run, as that’s going to be your denominator.

Send someone from your group up to the front whiteboard and report your results. Report the totals, not the experimental probability percentages. We’re going to calculate the results for the class.

The kids love the trials. They are religiously random, and really get a kick as they see a clear pattern emerge in the results.

After all the results are on the board, I tote them up and calculate the experimental probability for the class.

Then I transition to theoretical probability and discuss the difference between experimental probability—what actually happens in a series of trials—and theoretical, what is expected to happen. I ask for examples of trials that would have no theoretical probability: medical trials, new treatments, new procedures. I point out that researchers run thousands of trials because they want to have a reliable experimental model in order to begin to build theoretical probability.

In other cases—coin tosses, lottery tickets, and asking the Oracles’ advice—the theoretical probability is easily modeled. And that’s what we do next. Back to the handout, page 3.

Who Crosses the River?–Theoretical Probability

In Part B, you ran trials and calculated the experimental probability of each character’s being able to follow the Oracles’ advice. Now you’re going to determine the theoretical probability for each character and compare them.

Theoretical probability can be calculated when the number of possible outcomes is fixed.

In this case, we can define the following:

Sample Space:
The set of all possible outcomes for a trial.
Event:
Particular outcome(s) of the sample space. An event may contain other events, each with its own sample space.
Target event:
The desired outcome to be tested.

For example, the sample space for one instance of “asking for an Oracle’s advice” is: gryphon, Excalibur, dragon.

But in this case, we are asking for all the Oracles’ advice. So how do we find all the combinations possible from three Oracle requests?

There are three useful tools to help you model theoretical probabilities for a multiple-event scenario.

• A tree diagram can help you determine all the possible outcomes for a “compound” (multiple event) outcome, particularly complex events such as this one. Trees model a new “branch” for each component of an event.
• An area model is simple and easy, but limited to two or three events.
• A counting diagram is useful for ordering and calculating the number of outcomes.

See the flip side of this page for detailed descriptions of area models and probability trees.

Probability Tree

Working with your team, create a probability tree for all possible combinations of Oracular responses.

Using different colored pens for each character, trace the different outcomes: all three match (Vlad), two match (Dulcinea), no matches (Chaos).

Count up the possible crossings for each character, and the total possible outcomes.

How many different outcomes are possible? __________________

How many outcomes allow Vlad to cross? ________

Dulcinea? _______________

Chaos? _______________

Compare these numbers to your own experimental results, as well as the class totals. How do they compare?

I use a CPM handout (page 14), which isn’t all that great but gives visual examples of both models on one page. All I really want is the visual, which I haven’t gotten around to building for myself yet.

Building the probability tree diagram for “asking the Oracles” is a great activity that really brings home the difference between theoretical and experimental probability. The kids can see why Dulcinea gets the necessary agreement more often, and why Chaos wins more frequently than Vlad.

In earlier years I would have just had the students create the probability tree on paper, in their groups. But earlier in the school year, I came up with a fun way to work on bigger, multi-step problems. I have a lot of whiteboards. So in their groups, the kids go to a whiteboard section, and start working on the assigned problem(s). They have more room, I can see the work and be sure everyone’s got the correct answers. It shoves the math right under the nose of the weaker students, who might otherwise (ahem) sit quietly hoping I don’t notice they aren’t working or paying attention. And it mixes things up, which is always useful. I used this about three times during the year (our semester is a year); here’s a picture of them working on parabolas:

(Note: smudged the faces, took out the color, and asked the kids permission to use the photo.)

As I was driving to school the morning of this class, I suddenly realized that whiteboard work would be perfect for the probability trees. If I had them do it on paper, at least half of the kids would be looking on as someone else did all the work anyway, so I might as well be sure they were actually looking on, instead of tuning out. I usually use this method as review rather than for new concepts, but in this case it was the right call. Each of the groups were all involved in their own tree, and none of them were simply copying some other group’s work—some surreptitious checks to see if they were on the right track, sure. But that’s a bonus.

There were several gorgeous versions done in red, blue, green, and purple that I forgot to photograph before I erased them, but this one is nicely functional. Except the misspelling.

So we take a few minutes to compare the theoretical outcomes with the experimental, and since we have close to 200 trials (8 groups, 25 trials on average), they match up beautifully.

I tell them that science and research engage in experimental probability, but in math, we focus entirely on the theoretical by modeling the possible outcomes. I always start with the most flexible and visual, but the least useful, model. I then outline the other two models I want them to use, both of which I think have limitations, but are much more helpful: the area model and the “counting diagram”.

Counting Diagram

I always snicker at a formal name for a very simple concept. But it’s extremely useful as an organizer.

One blank line for every event. So asking the Oracles, there are three events.

______      ______      _______

How many outcomes are possible for each event? The events are independent of each other.

___3__        __3___        ___3___ = 27

Multiply across. That’s the total number of outcomes that can result in asking the Oracles.

Now, model each character. In these cases, the number of desired outcomes for subsequent events is conditional. (I point out that the Oracles’ response is still random and independent, but the desired response is conditional, and that’s what we’re counting).

Vlad is the easiest. The first event can be any of the three responses. But the second and third events must match the first, so there’s only one acceptable outcome for each.

___3__       __1___       ___1___ = 3

Dulcinea is more complicated, but the trees are very helpful in getting the kids to see that the first two events can have any outcome. The third event must match one of those first two.

___3__       __3___       ___2___ = 18

Finally, Chaos. Why is Chaos twice as likely as Vlad to get the correct Oracle response? The counting diagram helps students see that as each event occurs, he loses the possibility of that outcome—and yet, this gives him more outcomes than Vlad.

___3__        __2___         ___1___ = 6

The diagram can also calculate probabilities of multiple events, but it’s primarily useful for counting.

Right around now, I bring up lottery tickets, and we go through the hugeness of the numbers in a diagram. And here, I mention a key difference between theoretical and experimental probability, aka Why All Math Teachers Tell You Not To Buy Lottery Tickets.

Theoretical probability says it’s utterly pointless to buy lottery tickets. But every time the lottery runs, someone achieves the functional equivalent of getting struck by lightening while finding a four-leaf clover while getting abducted by aliens. Someone wins. Reality occurs. Gambling exists because of experimental probability. So my students won’t get the Lottery Tickets lecture from me. Go ahead and buy. Cross your fingers when your plane takes off, even though the car ride to the airport was the riskier trip. But if you blow your entire salary on poker, find a 12-step program.

Area model

My students are already

For this, I have a handout, because the area model makes the two most important probability operations beautifully clear: when to multiply probabilities, and when to add them.

Limitations—only two sample spaces, alas. You couldn’t model “asking the Oracles” with the rectangle.

So there’s the three basic models that we use for the rest of the unit. I often return to the “intern at the software shop” scenario, which gives me endless possibilities. Here’s a couple more.

This one is usually homework for the first day—what is the probability of getting various payouts? I don’t mean expected value—we go through that the next day, with the original Scintilla scenario and this one.

And here’s one that goes nicely with an area model, which can really help students visualize conditional probability.

Thanks to binomial expansion, probability and elementary combinatorics are sandwiched into second year algebra and it’s hard to go into the subject in depth. AP Stats is pretty joyless. Example 99,521,325 on the list of Why We Need to Offer a Broader Range of Math Classes.