Teaching Math: Experimental and Theoretical Probability

We can't predict the future, but-if all the outcomes are known-we can tell how likely it is that something will or will not happen. Students in upper elementary and middle school typically have their first experience understanding the difference between theoretical and experimental probability.

Definition
We use a number from 0 to 1 to tell the likelihood that something will or won't happen-this is called "probability." Probability can be written as a fraction, decimal or percent. The probability for an impossible event is 0 (you can't roll a 7 on a single, standard number cube), and the probability for a sure thing is 1 (the sun will rise every day.) The closer the probability gets to 0, the more unlikely it is that the event will occur. The closer the probability gets to 1, the more likely it is that the event will occur.

Theoretical vs. Experimental Probability
Theoretical probability is what should happen, in theory. If we flip a coin, we should get tails 50% of the time, or 1 out of 2 times.
Theoretical Probability: # of favorable outcomes / # of possible outcomes

Experimental probability is a result of trials that test predictions. We flip a coin 50 times to see how often it actually lands on tails. We must do a sufficient number of experiments.
Experimental Probability: # of favorable outcomes / # of trials

Discovery
Tell students that, at the end of the lesson, you're going to ask them to describe experimental and theoretical probability. Allow students to discover these probability terms; the concepts will then make more sense to them. These activities can be conducted over two or three days, depending upon the group.

Number Cube Activity
Place a number cube-or this simulated die-on the overhead, smart board or LCD projection. Ask the class what chance they think you have of rolling an odd number. Discuss why they think there is a 50% chance. Roll the die 5 times; record your results. Ask what they think will happen if you continue to roll 25 more times. Roll and record the results. The students will see that when more trials are conducted, the experimental results get closer to what should happen in theory.

Spinner Activity
Use this spinner from the National Library of Virtual Manipulatives. The spinner can be customized with different-sized regions, labels and so on. For example, you can create a spinner with 5 regions: 2 orange, 1 red, 1 blue and 1 green. Ask students how often the spinner will land on orange. Theoretically, P(orange) = 2/5 or 40%. Now conduct the experiment.The virtual spinner will record your results. You can spin one at a time, or set the spinner to quickly spin up to 999 times. Students will see how experimental probability eventually gets closer to theoretical probability.

Hands-On Activities
Place students into groups of 2-4. Distribute containers of manipulatives. Include items such as coins, spinners and multi-colored chips. Ask each group to create 4 probability trials. First, they should predict the outcomes, and then conduct experiments, recording the results. Showing them an example beforehand is a good idea. Afterward, each group chooses one trial to present to the class. Hang the results on a bulletin board.

Culmination
Ask students what they now think the terms experimental and theoretical probability mean. Ask them to write descriptions in their notebooks. Have several students share what they have written. Since they've discovered experimental and theoretical probability, these definitions should now make sense. This 4-minute video reviewing experimental and theoretical probability is a good culminating activity.

Paper and Pencil
Students invariably need to move to the abstract. Click here for worksheets and crossword puzzles that can be used as homework or follow-up activities. Answer keys are provided.

SKUNK Game
Students will have fun playing this game of chance.. Click here for an explanation and everything you need to play this game of chance and choice....the probability game, SKUNK. I played it as a whole class activity, but it can be played in smaller groups or pairs as well. Students love it.