Probability is an interesting topic. Really, it’s different from most mathematics our students learn. It’s the only topic in K-8 mathematics that doesn’t follow patterns. For example, we might know that flipping a coin 10 times should result in 5 heads and 5 tails, but in reality, it is quite likely that we will get some other result than this.

Because of this, I believe that we need to spend our time playing with tasks and making predictions, **a lot**, before we ever broach the concept of theoretical probability. We need to have students play the same games several times, where they can change their predictions based on previous experiments, before we provide opportunities for our students to understand the theoretical.

##### Big Ideas of Probability

When planning a problem in any topic it is always a good idea to consult Marian Small’s or Van de Walle’s “Big Ideas”. Here are 2 big ideas we looked at as we constructed the problem below:

- An experimental probability is based on past events, and a theoretical probability is based on analyzing what could happen. An experimental probability approaches a theoretical one when enough random samples are used. – John Van de Walle
- The relative frequency of outcomes (of experiments) can be used as an estimate of the probability of an event. The larger the number of trials, the better the estimate will be. – Marian Small

##### The Task Context

We explained to students that we had created a new game and wanted to test it out. While each group would play the same game, the board for each game was slightly different. We explained that it was their job to play the game and tell us which game board we should keep.

##### Rules for the Game

- The game is for 3 players
- 2 coins are flipped each turn
- Player 1 wins if 2 Heads are flipped
- Player 2 wins if 2 Tails are flipped
- Player 3 wins if one of each are flipped
- The result is placed on the game board (H for 2 Heads, T for 2 Tails, B for both – one of each)
- The winner of game is determined by the player with the most wins in that game.
- Once a game is finished the entire game is coloured 1 colour (Red for Heads, Blue for Tails, Green for Both). If a tie exists, 2 colours are used equally.

##### The Game Boards

Each student is given a game board with 100 sections, however, each individual game consists of a different number of trials.

Game board 1 – 1 game of 100 trials

Game board 2 – 2 games of 50 trials each

Game board 3 – 5 games of 20 trials each

Game board 4 – 10 games of 10 trials each

Game board 5 – 20 games of 5 trials each

Download copies of these game boards here

##### Results

After playing several games using each game board here is what we found:

##### What do you Notice? What do you Wonder?

Students noticed:

- The player who was chosen as “Both” won far more than the other two
- Game boards with more trials (game of 100 or 50 or 20) were coloured all/mostly green.
- Game boards with less trials (game of 10 or 5) had more red and blue sections than other boards, but still mostly green

Students wondered:

- Why did “Both” keep winning?
- Why is a game board with more trials more likely that green will win?

In the end, several students came to the conclusion that getting a Heads and a Tails must be more probable. Here is what they came up with:

##### Some things to think about

If a Standard/Expectation tells us that our students need to understand that experimental probability approaches theoretical probability with more trials, we can’t just tell students this. We need to set up situations where students are actually experimenting (including: making predictions, performing an experiment, adjusting predictions/making conjectures, re-testing the experiment…..).

As with anything in mathematics, our students need ample time and the right experiences to make sense of things before we rush to a summary of the learning.

I’d love to continue the conversation. Write a response, or send me a message on Twitter ( @markchubb3 ).

## 2 thoughts on “From Experimental to Theoretical Probability”