How Likely is That?

# 4. Take a Chance Suggested Learning Intentions

• To understand that a ‘game of chance’ involves chance rather than skill to determine the outcome
• To list the outcomes of chance experiments
• To define and compare experimental and theoretical probability

Sample Success Criteria

• I understand that a ‘game of chance’ involves chance rather than skills to determine an outcome
• I can list the outcomes of chance experiments
• I can compare theoretical probability and experimental probability
• I can record frequencies in a tally chart and calculate the relative frequency
• 1 coin
• Dice Differences - Prisoners game sheet: docx PDF
• 20 counters per pair of students
• 2 x 6-sided dice per pair of students

Play 'Heads and Tails' with your students. The game starts with all students standing. Informs the class that you will toss a fair coin and that each student must decide whether the coin will land on its head or its tail. Students make their selection by placing their hands on their head or hands on their tails.

After each flip of the coin, any student who predicted incorrectly sits down. The game continues until only one student remains.

At the beginning of the activity either ask a student to volunteer to record the result of each coin toss on the board or have students create a frequency table in their workbooks and have them record each outcome for the toss of the coin.  At the conclusion of the activity, students calculate the frequency and relative frequency of each outcome. As a class, discuss if the results of the experiment were what students expected. Ask students, ‘Did the experimental results match the expected results?’

Lead a classroom discussion around experimental and theoretical probability. Explain to students that theoretical probability describes what we expect to happen, whereas experimental probability is what actually happens when we try out an experiment. The more times we undertake the experiment the more closely the experimental probability and the theoretical probability are. For example, if we toss a coin 10 times it is unlikely that we will obtain a relative frequency of 0.5, however if we toss a coin 100 times or 1000 times, we are more likely to obtain an experimental probability close to 50%.

This activity is adapted from the maths300 activity called Dice Differences: Prisoners. It has been used with permission.

A fictional ‘prisoner release’ scenario is used as the context for this investigation. Please ensure that when you use this activity with your students it is adjusted as necessary to be contextually appropriate.

Provide students with a copy of the handout available in the Materials and texts section, or have them draw up a playing table in their books. Players begin by placing their ‘prisoners’ anywhere in six cells marked 0, 1, 2, 3, 4, 5. Each turn, two dice are rolled and one prisoner is released from the cell with the same number as the difference of the dice (assuming there are prisoners in that cell). The investigation involves students determining which ‘difference’ is most likely to be rolled, therefore releasing their prisoners in the least number of rolls.

Throughout the activity, students record each roll of the pair of dice in the blue segment of the sheet (experimental probability). At the conclusion of the activity, students calculate the experimental probability for each cell.

Next, ask students to write down all possible outcomes (theoretical probability) for each cell (in the orange section) then they calculate the theoretical probability for each. Have students compare the theoretical probability with the experimental probability.

Theoretical probability describes what we expect to happen, whereas experimental probability is what actually happens when we try out an experiment.

When running this activity in class, assign each student approximately 20 prisoners. Each prisoner is represented by a counter. Give students a number of counters which is not divisible by six. Students must make a conscious decision where they place their prisoners, rather than just sharing them equally amongst cells. You might want to introduce the rule that each cell must have at least one prisoner. This will prevent students from just placing all their prisoners in the one cell.

Enable students by providing them with two different coloured dice. This will help them see that 6 and 4 is different to 4 and 6 when recording all possible combinations to describe all possible outcomes.

### Areas for further exploration

1. Greedy Pig

This activity is adapted from the maths300 activity called Greedy Pig. It has been used with permission.

Students play this game as a whole class. The game involves 5 consecutive rolls of a six-sided die. Each time the die is rolled, the number which is rolled represents the amount of points won on that roll. For example, if a 4 is rolled, all students who are in the game win 4 points. However, if a 2 is rolled it makes everyone who is still in the game bankrupt, and they lose all their points. Students may decide to bank their total at any time by sitting down. After 5 rolls the game is finished. The student with the most money at the end of the game is the winner. Have students articulate why they sat down when they did. Was it after a certain number of rolls or was it after a certain number of points?

Run a second trial. Encourage students to make more mathematically informed decisions this time around.

At the conclusion of the second trial, extend students by asking them to collect each student’s grand total from both trials. Ask students to compare each set of results. What did they notice? Do students collect more points in the first trial or the second? Why do you think this might be?

Throughout this stage students develop their ability to calculate relative frequencies and compare theoretical and experimental probability.  At the conclusion of Dice Difference: Prisoners, ask students to write a reflection on the activity. Suggested reflection starters include:

• Which prison cell was the hardest or easiest to break prisoners free from? Why?
• If you were to play this game again, where would you place most of your prisoners? Where would you place the least?
• Which dice difference is the least likely to come up? Which dice difference is the most likely to come up?
• Can you use fractions, decimals, or percentages to describe the likelihood of each cell difference being rolled?
• Which cell has the most possible combinations of dice rolls?
• Compare the experimental probability with the theoretical probability. Did any of your cells exhibit a relative frequency that was the same as the theoretical probability?

Once students have reflected on the game, have them play a mini version in pairs. Encourage them to try out any new strategies they have found while reflecting.

Pairs Game

Each pair of students draws up the following six cell diagrams. Give each pair two dice and 6 counters. Students share their prisoners, placing no more than one prisoner per cell. Have students speed through the game. At the conclusion of the game, ask if their strategies were helpful. Do theoretical probabilities always align with experimental probabilities?

Australian Association of Mathematics Teachers, n.d. Greedy. [Online]
Available at: https://www.maths300.com/lessons?_token=qYckfQP1gvELp3DQ9GGauz0265zOLkdfzO4KFolx&q=greedy
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Australian Association of Mathematics Teachers, n.d. Prisoner. [Online]
Available at: https://www.maths300.com/lessons?_token=qYckfQP1gvELp3DQ9GGauz0265zOLkdfzO4KFolx&q=prisoner
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Freund, M. et al., 2019. The prevalence and correlates of gambling in secondary school students in Victoria, Australia, 2017. [Online]