Intent

Students learn how to use a random number generator to conduct a simulation of a probability situation. They are being prepared for the end of The Game of Pig unit, where they will use a programmed simulation that relies on a random number generator.

Mathematics

The theoretical probability of an event can be estimated by running repeated simulations. Technology such as a graphing calculator can be very useful in running many simulations that depend on randomness. It might seem to be a contradiction to use a machine algorithm to produce numbers that are supposed to be unpredictable. So-called random number generators actually produce what are called pseudorandom numbers. They aren’t random in the theoretical sense, but they are very close. [link to math maps]

Progression

The teacher introduces the idea of using a random number generator to conduct a simulation of the situation presented in Aunt Zena at the Fair. Students then design and use simulations to analyze experimentally Question 1.

Approximate Time

15 minutes

Classroom Organization

Pairs, followed by whole-class discussion

Doing the Activity

You might begin by asking the class, How might you simulate the situation in Question 1 of Aunt Zena at the Fair? Students might suggest something like putting 20 cubes in a bag, with one of them a different color from the others, and drawing out a cube to determine whether Zena wins or loses a given toss.

Explain that a graphing calculator has the capacity to do the same sort of thing. The calculator’s random number generator will pick a random decimal between 0 and 1. If possible, demonstrate how this is done using an overhead calculator. Demonstrate many results of the random number generator so students can get a feel for what is occurring. You might ask, What do you notice about what the calculator reports?

Now turn back to the situation at hand. How might you use a random number generator to simulate Aunt Zena’s situation? The most practical idea that students suggest will probably be something like, “If the number is .05 or less, she wins. If it’s more than .05, she loses.”

If students don’t raise the question of what to do if the random number is exactly .05, you might bring it up yourself. Ask what they think makes sense. One approach is to decide arbitrarily whether .05 will count as a win or a loss. If students are concerned that either choice will be unfair, you can ask what the chances are of getting exactly .05, assuming that the random number generator is truly random and is picking 10-digit decimals. Help them to see that this is a 1-in-10-billion occurrence, so they shouldn’t worry much about it.

Discussing and Debriefing the Activity

A class sample of about 600 trials (40 trials for each of 15 pairs of students) should give fairly reliable results. Combine the class results to see how close the fraction of simulated wins comes to the value of 120120 size 12{ { {1} over {"20"} } } {} given in the problem.

Once these data have been collected, you might ask about the advantages of using a random number generator for conducting a simulation compared with using dice or colored cubes. Do you prefer the concrete materials or the calculator? Students should be able to see that this electronic technique, while not as tangible, is more easily adjusted to different situations than pulling cubes from a bag or rolling a die.

Key Questions

How might you simulate the situation in Question 1 of Aunt Zena at the Fair ?

What do you notice about what the calculator reports?

How might you use a random number generator to simulate Aunt Zena’s situation?

Do you prefer the concrete materials or the calculator?