Introducing the Law of Large Numbers Through Simulation

We will allow students to understand the Law of Large Numbers through simulations, where they are looking for a certain outcome with the flipping of coins and the rolling of dice. 

Break students into groups of 2.

Supplies (for each group):

• Two standard dice
• One fair coin
• A calculator (students probably have these)
• Pencils/paper

Now a few simulations will be performed, but first students will predict the actual probabilities for the events they will be witnessing (somewhat like forming scientific hypotheses).

Simulation I: What is the probability of flipping heads on a coin once?

Have students calculate this probability.

1. Students should complete \(10\) trials. They record whether each flip is heads of tails. Then they find

$$\frac{number \; of \; heads}{number \; of \; flips}$$

This is the empirical probability, which we will denote \(E\).

2. Students should complete \(10\) more trials. They should again record whether each flip is heads or tails. They should find \(E\) cumulatively for all twenty trials.

3. Repeat step 2 with \(10\) more trials, with the final value of \(E\) encompassing all \(30\) trials.

Simulation II: What is the probability of getting at least one \(4\) when rolling two standard dice?

Have students calculate this probability.

1. Students should complete \(10\) trials. They record whether they got at least one \(4\) from each roll of two dice. Define this as a win. Then they find

$$\frac{number \; of \; wins}{number \; of \; trials}$$

This is the empirical probability, which we will denote \(E\).

2. Students should complete \(10\) more trials. They should again record whether they win or not. They should find \(E\) cumulatively for all twenty trials.

3. Repeat step 2 with \(10\) more trials, with the final value of \(E\) encompassing all \(30\) trials.

Students will likely observe the value of \(E\) get progressively closer to the actual calculated probability as they complete more trials. This is called the Law of Large Numbers and is the centralizing concept for this activity. 

Other tips:

• Consider having all students report their results on a table drawn on the board for sake of comparison. It is certainly possible in the first simulation that after \(10\) trials a group will have five heads, or \(10\) heads after \(20\) trials, and so on. Most groups will follow the suggested pattern for the Law of Large Numbers, however, and that is why comparison is important.
• If you have more than about 45 minutes, you can certainly throw in a third simulation in the same vein or, better yet, increase the number of trials done for each simulation to \(40\) or \(50\).