2. Theoretical and Experimental Probability

• A range of spinners showing differing probabilities
• Paper cups
• Cup toss recording sheet: docx PDF
• Frayer Model template: pptx PDF
• Dice

Facilitate a class discussion to revise key concepts of randomness and fairness from the previous stage (Probability of an Event). Introduce the terms 'theoretical probability' and 'experimental probability'. Explain that theoretical probability is what we expect to happen and experimental probability is what actually happens when we try it out, when we conduct an experiment.

Generate a set of statements with students such as “What is the chance that someone in our class has a birthday in January?” or “What are the chances of rolling a four on a die?” Determine the theoretical probability of each statement. For example, there is a 1 in 12 chance that a student in our class has a birthday in January.

Students can then gather data about the relative frequency of each statement and record their observations in a table and mark them on a probability scale. Facilitate a class discussion about any differences between expected and observed frequencies. 

Encourage students to consider the concept of theoretical probability further by displaying a range of spinners such as those provided in the example below. Use the Think, Pair, Share routine for students to consider, describe and explain their thinking using words and numbers about the theoretical probability of each spinner. 

Enable students requiring further support by directing their attention to the visual cues on the spinners to describe probability using words such as equal chance, less likely, more likely. What connections can they make to fractions, decimals and percentages?

Extend students by challenging them to come up with a generalisation (reasoning) for calculating the theoretical probability for any given situation. For example, the ‘formula’ for calculating theoretical probability can be described as 'theoretical probability equals number of outcomes in the event divided by the number of possible outcomes'.

• Can students prove that their generalisation works for all situations using their own examples?
• Are there exceptions to the rule (i.e., what happens when theoretical probability cannot be determined - this will be addressed in the Reflect and Consolidate phase).

Invite students to share their responses about how they described the theoretical probability of each spinner in words and numbers. Use talking strategies such as ‘Adding On’ (e.g., Would someone like to add on…) Repeating (Can someone explain/repeat what _______ said?) to sequence and connect students' responses in order to make explicit the connections between the ‘number of outcomes in an event to the number of possible outcomes.

Consolidate by sharing the online resource ‘Adjustable Spinners’ with students. Emphasise the difference between theoretical and experimental probability. Invite students to test their spinners, created in the previous stage, and record the expected probability and observed probability (relative frequency) in a table. Discuss unexpected outcomes. 

The following tasks are designed to draw student attention to the idea that sometimes probabilities cannot be determined by theoretical likelihood. Students will have the opportunity to collect, record and analyse data to determine experimental probability. Students will see how data is used in real-life to make generalisations about probability of events where theoretical probabilities cannot be determined. For example, theoretical probabilities cannot be determined for events like the chance of being struck by lightning, or the success of selecting players in a sports team based on their goal scoring average.

1. Which way up? A cup toss experiment

Display a paper cup (see handout in Materials and Texts). Ask students to consider the different ways the paper cup could land if you threw it in the air. Note there are three possibilities: right-side up (base), upside down, on its side. 

Ask students to predict which way they think it will be most likely and least likely to land and explain their reasoning. Students then test their hypothesis by tossing the cup 20 times and record how it lands.  

Ask students what they will need to consider so that they are consistent in how they toss the cup (e.g., same height) so that they can collate their results afterwards. Encourage students to systematically record their results in a table. Invite students to share their results and discuss differences between their predictions and observed results and suggest reasons for these differences. 

Ask students what they think will happen if they collate their data into one big table (moving toward an approximation of actual theoretical probability). 

In small groups (for example, five students per group to tally 100 tosses), ask students to pool their data and calculate the observed frequencies for each possible outcome. What do they notice? 

Repeat this process either by pairing groups together or as a whole class to determine the observed frequencies for 200, 300, 400, 500 etc tosses. Again, ask students what they notice as the pooled data grows? Invite students to revise their initial predictions. Do students feel more confident making predictions about the next 100 tosses? 

Support students to record experimental probability using fractions and percentages and make statements about their initial predictions. For example, if after 300 tosses, 238 landed on the side, you would feel more confident in predicting that a cup would land on its side approximately ⅘ or 80% of the time. This is called experimental probability. 

Enable students requiring further support by offering a pre-prepared table for students to record their observations (available in Materials and Texts).

2. Predicting what might happen in the future.

Pose the following scenario to students (adapt to suit interests of your class or if available, use students’ data if they can bring it to class).

Mim plays goal attack in the local netball team. The coach is trying to estimate the probability of Mim scoring 10 points or more in the next game. The coach collected data and created a graph that displayed the amount of points Mim scored in each game over the last two seasons. The graph looked like this: 

How would you advise the coach in working out the approximate probability of Mim scoring 10 or more points in the next game? 

Enable students requiring further support by asking students how many games Mim has played in total over the last two seasons? How can we connect this to our knowledge of fractions (i.e. denominator). You might also consider offering a reduced graph that contains only the first two columns and ask students to calculate the probability of scoring 5 or more points in the next game. The students then have a go at the main task. 

Extend students by having them explore different ways of writing and describing the probability. Ask them to make a judgement about whether this estimate is reliable. Challenge students to design their own question or provide data on personal bests from actual sports e.g., 100m sprint to select a relay team.

A follow-up task could include students exploring athletes’ track and field or swimming times (see, for example, Olympic Odds) to determine experimental probability and team selection. 

Areas for further exploration

• RIME Danger Fractions - similar to ‘Which way up?’ using a drawing pin.
• Take a Chance 
• Greedy Pig
• Dice Differences

'Which Way Up?' and 'Predicting what might happen in the future' exposes students to a range of probability concepts which you may wish to review and consolidate with students. Consider the progress made by students when determining which skills and concepts you select to review. 

Reflection on the tasks

Revisit the success criteria of this stage and invite students to add to their Connect, Extend, Challenge thinking tool :

• I can analyse a situation and determine what could happen/expected probability (theoretical)
• I can design and conduct an experiment, and collect and analyse data to determine how likely something will happen in the future/observed frequency (experimental)
• I can apply my understanding of fractions and percentages to express probability of events

Reflect on the task/s with the class. Invite students to discuss and write about the results that occurred in the experiment with 20, 100, 200 trials, for example. Discussion prompts include:

• Did the short-term and long-term results differ? How? Provide examples.
• Why did your initial predictions and the relative frequencies differ? Which is most reliable? Why?
• What might you do if the theoretical probability cannot be determined (i.e. estimate of actual probability)? Can you give examples of real-life situations when experimental probability is used rather than theoretical probability? For example, chances of getting struck by lightning, connect to Maxwell’s netball scores.
• Provide a question prompt that allows students to demonstrate their understandings of experimental probability. For example: How can we estimate what might happen in the future, based on experiments and/or what we’ve seen in the past?
• Demonstrate your understanding of key terminology of theoretical and experimental probability using a Frayer Model. A Frayer Model template is available in the Materials and texts section above.

Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Dice Differences. [Online]
Available at: http://mathematicscentre.com/taskcentre/034diced.htm
[Accessed 15 March 2022].

Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Greedy Pig. [Online]
Available at: http://mathematicscentre.com/taskcentre/200greed.htm
[Accessed 15 March 2022].

Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Take a Chance. [Online]
Available at: http://mathematicscentre.com/taskcentre/049takec.htm
[Accessed 15 March 2022].

Harvard Graduate School of Education, 2015. Project Zero: Think, Pair, Share. [Online]
Available at: http://pz.harvard.edu/sites/default/files/Think%20Pair%20Share.pdf
[Accessed 15 March 2022].

Harvard Graduate School of Education, 2019. Project Zero: Connect, Extend, Challenge. [Online]
Available at: https://pz.harvard.edu/sites/default/files/Connect%20Extend%20Challenge_0.pdf
[Accessed 15 March 2022].

National Council of Teachers of Mathematics, n.d. Adjustable Spinner. [Online]
Available at: https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Adjustable-Spinner/
[Accessed 15 March 2022].

Reys, R. E. et al., 2020. Helping Children Learn Mathematics. Milton: John Wiley & Sons Australia..

Russo, T., 2020. The King's Tax Maths Game. [Online]
Available at: https://www.youtube.com/watch?v=kAcBuOXiLwk
[Accessed 15 March 2022].

Siemon, D. et al., 2015. Teaching Mathematics: Foundations to Middle Years. Melbourne: Oxford University Press.

State Government of Victoria (Department of Education and Training), 2019. The Frayer model. [Online]
Available at: https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/english/literacy/Pages/lim_terminology.aspx#link2
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. Adjustable spinner probability activity. [Online]
Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=2c0ad2ec-2863-47cf-b780-0c6414ef992f
[Accessed 15 March 2022].

State of Victoria (Department of Education and Training), n.d. Danger Fractions: a mathematics problem solving RIME task. [Online]
Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=10c9bfa5-c138-4540-a43c-11cf48ae4f45
[Accessed 15 March 2022].

Swim Swam, 2020. Olympic Odds: each swimmer's probabilty of winning gold in Tokyo. [Online]
Available at: https://swimswam.com/olympic-odds-each-swimmers-probability-of-winning-gold-in-tokyo/
[Accessed 15 March 2022].

Van de Walle, J. A., Karp, K. S. & Bay-Williams, J. M., 2017. Elementary and Middle School Mathematics: Teaching Developmentally. Ninth ed. Harlow`(Essex): Pearson Education Limited.