The Human STELLAã

An introduction to STELLAã Modeling

The diagram below illustrates the model of a simple binomial game (see http://davidmlane.com/hyperstat/A2301.html for an explanation of binomial events) that the students will work with in their next activity. In order to help them understand the model, we will have them physically perform the actions of the different components of the model. Material needs will consist of three containers (plastic bowls will work), twenty to thirty objects to be used as counters and moved from container to container (marbles, blocks, ???) and a randomizing device that will allow the student who is RANDOM GEN to determine his result (anything from a coin, die, or random generator on a calculator or computer), and two 5x8 index cards with a large “1” on one of them, and a “0” on the other.

Instructions.

Explain the purpose of the activity and reproduce the illustration so students may have an overview of the process. You may present this as a model of drawing marbles from a box in which a certain percent of the marbles will win a prize, and the rest will not.

Using the model as an illustration, set out containers and identify them as the containers for TRIALS, Wins, and loosses.

Place the 20 or so “marbles” in the Trials container. Explain that we will make all the marbles indistinguishable for now, and identify the type as we draw them; this makes the model more flexible.

Now we need to create an object that will decide what type of marble (winner or loser) that each marble is as they are drawn. To do this we select a student who will serve as the RANDOM GEN. This student has the responsibility of declaring each marble as a win (1) or a loss (0). To accomplish this, they will need a randomizing device. We will use the idea of a single die as the random device for these instructions. Before we begin we need to establish what proportion of the marbles should win, and since we are randomizing with a single die, the choices must be in sixths. We place the proportion of success in the controller marked “prob wins”. It should be labeled and written on the board. If “prob wins” is 1/3, for example, then RANDOM GEN will declare the result by holding up the appropriate of the two index cards; a winner(1) if he rolls a one or two on the die, and a loss (0) if he rolls anything else. In general, the purpose of Rand Gen is to output a 1 if the random device produces a result with a probability LESS than the value in “prob wins” and a zero otherwise.

Next we select two students to represent the pumps that move trial values into the WINS or LOSSES bins. The student representing the win pump will watch the output of RANDOM and if it is a win(1) he will move one marble from the trial bin into the WINS bin. The pump for the LOSSES bin will observe the RANDOM response, and move one marble to the LOSSES bin if a loss (0) is indicated.

Two more students may be selected who will represent the displays for the number of wins or losses. They will write on the board the current count of marbles in the WINS or LOSSES bins for which they are responsible to report.

Each time the teacher indicates a new iteration of the process will take place in the following sequence:

1. The RANDOM GEN will activate his randomizing device and report a 1 or a 0 with the appropriate outcome card according to the random event.

2. The pumps will move or not move a counter from the trials bin to their indicated receptacle.

3. The reporters will update their displays.

At this point they await the instructions to repeat the process again.

The complete process should be run more than once to allow students to see that even with the same inputs, the results depend on the chance events during the experiment. Discussions should include questions that insure that students understand that the expected number of successes depends on the probability of success on each trial, and the number of trials. It is also suggested that students have an opportunity to discuss how the model could be altered to simulate different situations.