What are the Chances?

# 1. Probability of an Event

Suggested Learning Intentions

• To describe the probability of an event using words and numbers
• To understand the concepts of randomness and fairness and their impact on determining likelihood of events

Sample Success Criteria

• I can represent probability on a continuum from 0-1
• I can use the language of chance: certain, likely, equal chance, unlikely, impossible
• I can express probabilities using fractions, decimals, and percentages
• I can describe whether the probability of an event is random and/or fair
• I can use manipulatives to model my solution and thinking
• Rope/twine/yarn
• Coloured counters/unifix blocks
• Dice
• Fraction, decimal and percentage cards: docx PDF
• Spinner template
• Testing your bag design - record sheet: PDF

The following activities are designed for students to develop a clear understanding of the language of probability. These tasks also prompt students to assign, debate and discuss the associated numerical values on a probability scale. Consider creating a modified Connect, Extend, Challenge thinking tool (either as a whole class or for individual students), which can be revisited and added to as appropriate throughout each stage of the sequence.

Share the following newspaper article with students: Europe | Bulgarian lottery repeat probed. Summarise the article:

In Bulgaria, the lottery uses the numbers 1 to 49. On 6 September 2009, the winning numbers were 4, 15, 23, 24, 35 and 43. In the following draw on 10 September 2009, the exact same numbers were drawn. This led to an investigation to make sure that no one had ‘fixed’ the game.

A mathematician concluded that there was a ‘one in four million’ chance of picking the same six numbers twice in a row and that even the most unlikely results can happen. Actually, the probability of selecting the six winning numbers in any single lottery with numbers 1 to 49 is approximately 1 in 14 million.

Display a long piece of rope on the classroom floor. Place the numbers ‘0’ and ‘1’ at either end of the line. Distribute the fraction, percentages and decimals cards and invite students to place them along the line.

Ask students to share examples of events that match the different likelihoods on the line. Offer sticky notes to students to record their example and place it on the probability scale. Encourage students to consider the appropriateness of their placement with their peers and consider the following: Are ‘very unlikely’ situations still possible and can ‘very likely’ situations not happen?

Discuss what students notice about different ways of describing probabilities using fractions, decimals, percentages, and words. What do students consider when describing probabilities? What is more useful? Why?

Enable students requiring further support by providing a set of spinners for students match a visual representation along the probability continuum. Ask (for example): "Can you match the probability of spinning blue to 50% chance? What could be an event that would match that probability? How do you know?"

Extend students by considering additional ways of expressing probabilities using fractions, decimals and percentages and associate language. For example, ¹/₁₀, 0.1, 10%; ¹/₃, 0.333, 33%. Challenge students to come up with probabilities using equivalent fractions and describe events to match the probabilities. For example, 3 in 12 chance is the same as 1 in 4 and is the same as 25%.

The following activities provide multiple exposures for students to work in small groups to describe probabilities using numbers and words and represent these on a probability continuum. As a precursor to the next stage (Theoretical and Experimental Probability), these tasks are also intended to build connections between the concept of randomness, fairness and likelihood of events.

1. Concept of randomness - Are you a winner? (Reproduced from Reys et al, p. 648).

For something to be random, it cannot be influenced by any other factor than chance.

Our class is having a lucky dip. Each person places his or her name in the draw just once. One name will be randomly picked, and that person will be the winner. The names are placed back in the draw after each turn and the lucky dip is shaken up.

Read each of the following statements and think about the people in our class. Decide where the following statements should be placed on the probability scale below.

• The winner will be left-handed.
• The winner will have brown hair.
• The winner will be someone in our class.
• The number of letters in the first name of the winner will be less than the number of letters in their last name.
• The winner’s first name will begin with a vowel.
• The winner will be wearing a watch.
• The winner will be wearing socks.
• You will not be the winner.

Questions to consider as a class or small groups (Reys et al., p. 647)

• Why is it important that the name be randomly picked?
• Should the names be seen by the person doing the draw?
• Would it matter if some people wrote their names of large pieces of paper and others on small pieces of paper? (Leads to the idea of fair/unfair).

• How might you adapt the lucky dip so it’s more likely for you to win than your classmates?
• Is the game random? Why or why not?
• Is the game fair or unfair? How do you know?

Ask students to design a game (using dice, cards, or spinners) that is either fair or unfair. In other words, the players have an equal chance of winning, or one player has an unfair advantage over the other/s. Is the game ‘rigged’? Use mathematics to justify your thinking.

2a. Designing random devices: Pulls from a Bag

View the Expressing probability mini-clip in ClickView (login using your department credentials).

Facilitate a class discussion on the key concepts contained in the video. For example, the following discussion starters might be helpful:

• Is the chance of pulling out a green pencil from the pencil case random? Why/why not?
• What colour cat do you think the child will pick with their eyes closed? What would we need to change to make it fair for all the cats to have an equal chance of being picked? Use words and numbers to justify your thinking.

Designing a bag

Explain to students that they are going to design a bag to test their understanding of randomness, chance and probability. Students will need to decide how many counters and how many colours they will include. Consider starting off with ten or twelve counters and limiting the number of colours for enable students who need support.

Provide students with an example ‘bag template’. For example, 'My bag has 12 counters. Of those 12 counters, 6 will be red, 4 yellow, 2 blue.'

Have students mark the probability of pulling each colour out of the bag on a probability scale and represent it using fractions, decimal and percentages.

Next, have students design their own bags. Encourage students to choose several criteria and select counters to match those. For example, a student may want to have the criteria ‘four blue counters’ and ‘50% green counters’. They could achieve that with a bag that had four blue counters, six green counters and two yellow counters.

Ensure students record and explain their thinking about their selection of colours and associated probabilities. For example, 'I predict that I will pull a red counter 50% of the time because ⁶/₁₂ is the same as ½ or 0.5.'

Enable students by reducing the number of items in the bag to 10 or 12 and make connections to fractions decimals and percentages (¹/₁₀, 0.1, 10%). You might also reduce the number of colours in the bag to two.

Extend students by changing the number of items they include in their bags (for example: 15, 25, 50) to work with other fractions, decimal and percentages (for example, 1 in 15 chance). Challenge students to come up with probabilities using equivalent fractions and describe events to match the probabilities.

Observe students’ reasoning in how they design their bag and estimated chance. How do students perceive chance as appearing on a continuum?

• Encourage students to share their reasoning for their bag designs and the probability of pulling out each colour.
• Ask students what they think will happen if they pull out a counter and replace it ten times. Do they think their predicted probabilities will match what they get? Why/why not?

Testing probabilities

• Students test their bag by pulling out a counter (and replacing it) 10 times and recording what colour they pulled out. Ask students to repeat the trial at least three to five times.
• Ensure students are systematically recording their observations using a table such as in the example provided below. A template is available from the Materials and texts section above.

• Ask students to tally their results according to each colour they pulled. Encourage students to calculate the percentage of occurrences of each colour that was pulled.

Comparing predictions to observed probabilities

• Ask students what they notice about their predictions and what is observed.
• How did their trials turn out?
• Did they match their predictions on the probability line?
• Were there surprise results?
• Why is it important to conduct repeated trials?
• Pose the following statement ‘Chance has no memory’ on the board and ask students to turn and talk. What do they think this means?

### Areas for further exploration

Designing random devices: Spins on a spinner (revisited and elaborated on in the next stage)

Tell students:

'I spun a spinner many times. It landed on green most of the time, on blue some of the time and once on red and yellow. What might my spinner look like? Use words and numbers to describe the probability of landing on each colour.'

Designing random devices: Coin toss

Tell students:

'Saahm flipped a coin 6 times. It landed on heads 6 times. Saahm flipped the coin again. What is Saahm likely to flip the seventh time? Why? Explain your answer in two different ways.'

(HINT: think about the concept of randomness and why events are equally likely).

Additional activities, available from the Mathematics Centre and NRICH

This stage has focused on consolidating students’ understanding of describing probabilities using words and numbers. Students should demonstrate an awareness of the concepts of randomness, fairness, and variables that affect the probability of events.

Facilitate class discussions to elicit students’ thinking about the concepts of probability, randomness and fairness. Listen to students' responses to assess the degree to which students have shifted their thinking from a belief in pure chance and luck, to understanding that irrespective of luck, some results are more or less likely to happen.

Check for student understanding of these key ideas by inviting students to investigate the following game and argue mathematically as to whether the game is random, fair, or unfair.

Fair or Unfair? A Coin Toss Game - 3 players

Two coins are tossed. If both coins land with heads face up, Player A wins. If both coins land with tails face up, Player B wins. If one coin lands on heads and the other coin lands on tails, Player C wins. Is the game fair or not? Why? Use mathematics to explain your thinking.

• Predict what you think will happen.
• In groups of 3, play the game 20 times and record the outcomes. How do the results of this small sample compare with your predictions?
• Argue mathematically if the game is fair or not, and the chance of winning the game. Support your thinking by identifying each player’s chances of winning on the probability scale.
• Write a definition of a fair game. For example, a fair game is when everyone has an equal chance of winning.

Enable students requiring further support by asking them to list what the possible outcomes are by using two coins (i.e. HH, TT, HT, TH).

Extend students by asking them to draw a diagram to find all the possible solutions and the chance of them occurring.

Monitor students’ responses. For example, how do students analyse the situation (the game) without playing? Do they make accurate predictions? Do they notice that there are two ways for a head and a tail to occur? Or do they need to play the game to help them understand why the game is not fair? Do students recognise that each toss is independent of one another? These ideas will form the basis of formally introducing theoretical and experimental probability in the next stage.

Invite students to add to their Connect, Extend, Challenge thinking tool to monitor shifts in their thinking.

BBC News, 2009. Bulgarian lottery repeat probed. [Online]
Available at: http://news.bbc.co.uk/2/hi/8259801.stm
[Accessed 15 March 2022].

Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Cat & Mouse. [Online]
[Accessed 15 March 2022].

Black Douglas Professional Education Services, n.d. Mathematics Task Centre: Win at the fair. [Online]
[Accessed 15 March 2022].

ClickView, 2016. Expressing Probability. [Online]
Available at: https://online.clickview.com.au/share?sharecode=882325b0
[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].

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. Creating texts: Word and sentence level. [Online]
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. Probability: certain, impossible or likely. [Online]
Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=2c2f1b67-32cd-482d-82d5-ff7aa7cfcd90&SearchScope=All
[Accessed 15 March 2022].

Toy Theatre, n.d. Marble Jar. [Online]
Available at: http://toytheater.com/marble-jar/
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Misunderstanding Randomness. [Online]
Available at: https://nrich.maths.org/6107
[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.