3. Sample Space and Independence of Events

• dice
• deck of cards
• Probability table using cards: docx PDF

This stage provides students with opportunities to connect previously explored ideas about fairness, theoretical and experimental probability. It also builds on their understanding to explicitly explore the probability concepts of sample space, visual representations, and independence of events. Students will also design, conduct and analyse simulations and experiments that form the basis of the next stage.

Introduce the concept of sample space by revisiting 'Fair or Unfair? A Coin Toss Game' from Stage 1. Ask students what the possible outcomes were for the game and display this on the board. For example:

Explain that this is called sample space: the set of all possible outcomes for a chance experiment. Facilitate discussion to connect students’ thinking to theoretical probability.

Ask:

How does representing the sample space assist us to determine theoretical probability? You might also consider introducing the idea of independence of events: that two events are independent when one event does not affect the outcome of the other. 

You might also use this opportunity to compare the use of a table to a tree diagram to represent sample space. Ask students to compare what is the same and different. Which do they prefer? Why?

View the ClickView resource Probability Basics (log into ClickView using your department credentials) and follow up with a discussion about how the presenter and Sunny explore probability concepts such as sample space, favourable outcomes, possible outcomes, theoretical probability and experimental probability. Also revisit the way the presenter and Sunny describe the various probabilities using fractions, decimals, and percentages.

Support students to explore these ideas further by posing questions about determining theoretical probability using fractions, decimals and percentages from a given sample space. For example, how could we represent the sample space of a deck of cards? Encourage students to think about how they could represent this collaboratively.

For example, you could use a table below and ask:

• What is the probability of drawing a five from a deck of cards?
• What is the probability of drawing a heart from a deck of cards?
• What is the probability of drawing a royal from a deck of cards?
• What is the probability of drawing a black card from a deck of cards?

Provide a recording table for students to document and track their responses. Ask students:

• What stays the same and what changes?
• Would we describe this as an independent or dependent event?
• What would need to happen to turn it into a dependent event?
• How might a table help to document and track our thinking?

Students generate their own examples. Consider revisiting students’ “Pulls from a bag” designs in Stage 1 to record the sample spaces and describe independence of events. It would also be worthwhile to revisit their results to focus on short-term variability and long-term stability. For example, ask students “how did your short-term results compare to the observed probability when you conducted more trials?”

2a. Exploring independence of events with dice. 

The following task gives students the opportunity to explore a variety of probability concepts including sample space, independence of events, short-term variability, and long-term stability, and determining experimental probability. It also provides students an opportunity to practice basic facts and examine properties of numbers. 

Distribute and display a six-sided dice. Ask:

• When I roll this die, what numbers could I land on? (These are called the 'possibilities'. The set of all possibilities is called the 'sample space'.)
• What is the probability of rolling a 6?
• Are all possibilities equally likely? Why? How do you know?

After these questions, re-introduce the sample space.

We already discussed the sample space of rolling one die. What if we bring in another die, so now we have two? What is the sample space now?

Introduce a second die and explain to students that, in pairs, they are going to roll and record the sum of both dice in the following table (it will appear as a bar graph) until one of the columns is full. See the below image for a completed example, where the sum '7' was rolled 10 times.

Before students complete the task, ask them to 'turn and talk' or use the donut share classroom talk technique to discuss, record and explain their predictions about the following:

• Which sum will they reach to ten first?
• Which sum will they roll the least?

While playing, it might be useful to ask students to highlight which sum they got the first in each row. For example, If a sum of 3 was the first to occur once (i.e. the first roll), then you would highlight that number. If a sum of 7 was the first to occur twice, you would highlight the number in the second row.

After students have played, discuss which sums won the most and the least and possible explanations. Pool students' data for each of the sums rolled the first time in each of the rows and support students to compare these to their results. Encourage students to examine the ‘shape’ (variability) of their data displays (table) and how it changes as the number of games increases. Which numbers continue to win the most towards the end of the game? Why might this happen? 

Which sum do they think would get first to ten if they played again? Why? It would be useful to get small groups of students to pool their data to examine how the variability gets smaller and why this might happen. Revisit the idea of theoretical and experimental probability.

2b. Using sample space to determine probability

Another way for students to record their results from 2a. is using a two-way tally chart. You might decide to repeat 2a using a tally to record their results in each cell of the table below. To predict and explain the results in 2a, students can represent the sample space for both independent events for the sum of rolling two dice in a table.

Use the following prompts to encourage students to explore the sample space:

• What probability statements can you make based on the sample space? For example: the probability that the sum is not 6 is __________. The probability that the sum is less than or equal to 9 is ___________. Use fractions, decimals, and percentages.
• Based on the sample space, what do you predict to be the most and least likely sum? Why?
• Is it more or less likely to roll an even or odd sum? How do you know?

Enable students requiring further support by linking the total outcomes to the denominator (for example, the probability of rolling a sum of 2 is ¹/₃₆). How many combinations of each sum are there? What is the probability in fractions, decimals, and percentages? What other connections can students make?

Extend students by asking them to imagine they multiplied the numbers rather than adding them. Ask students to record the sample space of results when two dice are multiplied. Is the product more likely or less likely to be odd or even? Why? Prove your thinking using mathematics. What do students notice about the products of odd and odd or odd and even numbers?

1. Rock, Paper, Scissors, Lizard, Spock (RPSLS)

The following has been adapted from reSolve: Maths by Inquiry (Probability: Rock Paper Scissors) and the Maths Curriculum Companion (See Teaching Ideas: Rock Paper Scissors) and is an opportunity for students to connect, apply and consolidate previously explored probability concepts.  

Invite students to play a game of Rock, Paper, Scissors. Facilitate a discussion about who is more likely to win and why? Is it a fair game? Why/why not? What is the likelihood of winning/drawing?

How can we use sample space to help our predictions? Invite students to represent the sample space in their preferred way. 

Introduce a modified game of RPS by watching The Big Bang Theory - Rock, Paper, Scissors, Lizard, Spock (RPSLS). In the clip, Sheldon claims in the traditional version of RPS, ‘players familiar with each other will tie 75% to 80% of the time.’ Discuss this claim with students. Do students agree/disagree with Sheldon? Why/why not?

Ask students: Do you think your chances of winning increase, decrease or stay the same if we play RPSLS? How can we find out?

• Encourage students to visually represent their thinking using a table or tree diagram to justify their chances of winning the game. Describe the probabilities using fractions, decimals and percentages.

• Monitor students’ responses: do students notice that the chances of a tie decreases and the chances of winning and losing increases?
• Student can play the game and compare theoretical and experimental probabilities and discuss ideas of fairness and randomness

Enable students requiring further support by asking to think about the amount of moves that can be made (5), and the amount of ways you can win (2), lose (2), or draw. 

Extend students by asking to design new moves for a game with some parameters. For example, a game needs a minimum of three moves to be certain that a move either wins or loses. What new moves can students make so that:

• With three moves, the chance of winning is 3 in 9 or one third
• With five moves, the chance of winning is 10 in 25 or two fifths
• What would be the chance of winning with seven moves? Nine moves? 101 moves? 
• Can you make a generalised statement to determine the chances of winning with any odd number of odd moves?

Areas for further exploration

1. Mystery lunch order

Lin volunteers in the school tuckshop to help prepare lunch orders. Lin accidentally forgot to label the last three lunch orders. In each bag Lin put:

• one of two sandwiches (cheese and vegemite, or salad)
• one piece of three pieces fruit (apple, banana, or an orange)
• one of two snacks (carrot cake, or gingerbread person)

What is the likelihood that the lunch order Rami gets has a salad sandwich and a gingerbread person?

Use mathematics to represent and explain your thinking.

2. In a hurry

In my closet, I have three pairs of pants (jeans, tracksuit, leggings) and four t-shirts (white, black, grey, and blue). I also have two pairs of shoes (runners and thongs). I was running late one morning, and I quickly grabbed the first things I saw and quickly got dressed. What is the probability that I randomly picked jeans, a grey t-shirt, and runners? How do you know? Use mathematics to represent and explain your thinking.

3. More ideas

• Duelling Dice 
• Crazy Animals
• Choosing beads
• Have a Hexagon 
• Sicherman Dice 

Pose the following open-ended problem to students:

• A sample space for an experiment has 12 outcomes. What might the experiment be about? How might you represent the solution?

Monitor students approaches. Do students:

• create simple one-way or more complex two-way experiments (flipping two coins)?
• describe the sample space for experiments rather than simply defining a sample space for a one-stage or two-stage experiment?
• notice independence of events? 
• effectively use a table/matrix, tree diagram, or area model to illustrate a sample space? Which diagram do they use to determine the sample spaces and probabilities? 
• use diagrams with understanding?
• represent probabilities using words and numbers?

Enable students by reducing the number of outcomes (such as four). Alternatively, provide an example that students can make inferences about. For example, you could adapt the ‘Mystery lunch order’ task.

Extend students by challenging them to develop a plan for designing and conducting an experiment.

It is not necessary for students to conduct the experiment if there is no time. Instead, you can ask for written and verbal explanations for each decision made in designing the experiment. Teacher prompts include:

• What do you want to find out? What is your question you want to explore?
• Does your model for the real-life situation make sense? How? Why?
• How will you set up your experiment? What do you need to consider?
• How many trials will/did you do? Is that sufficient? Why/why not?
• What is your estimated probability? Does it make sense? Why/why not? 

It is important for students to develop their understanding about the difference between short-term variability and long-term stability. Facilitate class discussion about the law of large numbers and monitor students’ responses: What does it mean when someone says….

• “In the long run…” 
• “It all works out in the wash…”
• “Even Stevens”

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

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Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Choosing Beads. [Online]
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Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Crazy Animals. [Online]
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Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Duelling Dice. [Online]
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Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Have a hexagon. [Online]
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Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Sicherman Dice. [Online]
Available at: http://mathematicscentre.com/taskcentre/241sichr.htm
[Accessed 15 March 2022].

Clickview, 2011. Probability Basics. [Online]
Available at: https://clickv.ie/w/AGTl
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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].

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

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

State Government of Victoria (Department of Education and Training), 2020. Literacy Teaching Toolkit: Classroom Talk Techniques. [Online]
Available at: https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/english/literacy/speakinglistening/Pages/exampleclasstalk.aspx
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. Describe probabilities using fractions, decimals and percentages. [Online]
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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.

Warner Bros. TV, 2013. The Big Bang Theory -- Rock, Paper, Scissors, Lizard, Spock. [Online]
Available at: https://www.youtube.com/watch?v=iSHPVCBsnLw
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