Year 9 students investigate probability in a range of contexts, assigning probabilities to outcomes and determining probabilities for events.
Probability, Dependent Events, Independent Events, Complementary Events, Experimental, Theoretical, Scientific Notation, Tree Diagrams, Fractions, Decimals, Percentages, Multimedia.
This Learning Module is based upon the Australian Curriculum: Statistics and Probability.
As a result of completing this Learning Module, students will be able to:
Learning Intention: To understand what probability is.
Form small groups with a maximum of 6 students in each. Each group will need one pad of small post-it notes. Your group has 3 minutes to write any and all words or ideas that you can think of that relate to the words “Probability”. Only one response should be recorded on each post-it note. Your group should aim for at least 5 responses.
One member of your group should be selected to place each of your responses into the appropriate category written by the teacher on the whiteboard.
Comment: As a class look at your different responses on the whiteboard and think about what sort of things you know about probability. What sort of responses did other groups have that yours didn’t? Did anyone know any maths related to probability? What words came up a lot? What sort of examples did people know? Build on other students' comments by starting with @Name.
Probability Mind Map
[You will need: Small post-it notes]
This activity is designed to find out what students already know and understand about Probability. It should be completed in small groups (maximum of 6) to allow some discussion of ideas. Each group brainstorms any and all words or ideas they can think of that relate to the word “Probability”. If students are having a hard time you may wish to point out that the word is derived from “probable” or lead them to the word “chance” and see if this helps. Each response should be recorded on a small post-it note and saved for use later.
As the students are brainstorming the teacher should create a graph on the board that has a column for each of the main ideas and concepts that link to probability (e.g. language (words related to likelihood or chance), ways to represent probability (percent/decimal), specific real world examples (lotto), Maths (facts/equations/concepts/definitions)).
Ask the students to select a member of each group to come to the board place their post-it’s into the column that they think they belong. You will need to first describe what each column represents and may need to help guide the students to correctly place their responses.
Guide a class discussion on the nature of their existing knowledge – what language did they use? Were there 1 or 2 particular words that came up a lot? Did they have any specific mathematical knowledge of the topic? Were their lots of specific examples (different examples or the same one’s repeated?)
Record the class’s responses by either taking a photo of the post-it chart on the whiteboard and posting it in an Update in Scholar for students to comment on.
The aim of this Update is to introduce the Scholar environment to participants and to value their prior knowledge. Working in Scholar promotes:
Learning Intention: To explore probability through a game of chance.
You are going to play a simple dice game called ‘Pig’. This game is played as a whole class but you will each get your own score which you will need to keep track of on your scorecard. All students need to stand up to begin play. A six-sided die is rolled. The number on the die is your score and you need to write it down for round one. You can continue to write down the scores for each roll of the die so long as you are standing. However, if the die lands on a 1, then everyone still standing has their score set to zero and that round is over. You can sit down whenever you think a 1 might be the next roll. At the end of the six rounds, sum your scores. The student at the end with the highest score is the winner.
Comment: What did you learn about probability from playing 'Pig'? Comment on other students' comments, building on their ideas and giving positive feedback to them.
Probability Game
[You will need: A die]
This simple game, Pig, draws on students' intuitive understanding of probability-based concepts that they have developed without being explicitly taught. If you play this game at the start and end of the Learning Module you will hopefully see a change in outcomes resulting from a deeper understanding of probability.
Notice how many rolls of the die the students remain standing. Make sure the students keep their score sheets for comparison at the end of the unit when this game is repeated.
Learning Intention: To find out where probability occurs in our lives.
Watch the clips from Fight Club, 21 and Bangkok Insurance.
Complete the Probability and Game Shows worksheet.
Comment: What did you learn about probability? Share your ideas and then comment on other students' comments, building on their ideas, and noting similarities and differences.
Probability in the Real World
Learning Intention: To define probability.
Complete the Sum to 7 Probability Worksheet.
Comment: What are some important things to include in a definition of probability? Keep adding to people's ideas so there are many ideas that you can draw on for your Video project in the next Update.
Defining Probability
[You will need: Sum to 7 handouts]
Each student is to write their own definition of probability. They will then compare their definition with another student, These two students must give each other a score which totals to 7 using only whole number.
Without changing their definition, each student needs to meet up with another 4 students (5 in total) in order to compare definitions and give each other scores.
At the end of this, ask for students to share who had the greatest score out of a possible 35.
Have the top 5-10 students share their definitions, writing key words on the board from the definitions as you go.
Then, as a group, determine a whole class definition for probability.
The Video Project will be assigned in the next activity. This Update will support students to complete Scene 1 of their assignments.
Learning Intention: To start your Video Project in Scholar.
In a pair, create a short video that will explain the ideas and concepts you have learnt while studying the unit on Probability. As you proceed through the unit you will be asked to complete sections of this project that relate to each new concept covered. You can add your notes to sections in the Structure Tool in Creator in Scholar. You can then elebaorate on these to create your final video for submission.
Requirements:
Check your Notifications for a "Work Request". This will take you into Scholar where you can start your work. Look at the Rubric to see what is expected of you. Use the Structure Tool to create 5 sections for your work. Start with Scene 1: Define Probability. Each student needs to create their own notes and script, even though you are working in a group, and producing one video.
Comment: Do you have any questions about how Scholar works? Make a comment in this update. If you think you have an answer to another student's question, please answer it - be sure to name the student you are replying to in your comment by starting with @Name.
Ongoing Group Video Project - Major Assessment Task
As part of the assessment for this Learning Module, groups of students will create videos in which they explain each of the major concepts covered and then discuss a topic involving probability that affects society.
Every student should add to their project in Scholar as they progress through the Learning Module, even though they are creating one video together.
In class or for homework, have the students discuss and complete the work on their first scene where they will provide a definition of probability.
You may choose to assign each group their social issue now or later. Please refer to Update 19 for more information.
When setting up the project settings in Scholar, manually assign one reviewer so that you can ensure that students review the work of a person in another group - not their own.
Ensure that students look at the rubric so they have clear understanding of the assessment criteria.
Project Rubric
Learning Intention: To match the language of probability with a number scale.
Look at each of the listed events and decide what chance they have of occurring:
The events you are to consider are:
You should now take your list of events and place them on the scale shown in order from impossible to certain.
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| Impossible | Highly unlikely | Unlikely | Even chance | Likely | Highly Likely | Certain |
If we define the probability of an impossible event to be 0 and a certain event to be 1 work out an approximate number scale to place on the scale shown above.
Comment: Using the language of probability, describe the probability of an event. Make a class list with everyone contributing. Make sure you do not repeat what another student has described. You can create different events than the ones in the list too. Make suggestions for alternative ways of describing probability of events on other students' comments. Start your suggestion with @Na
Understanding the Language of Probability
In this activity students will begin to consider probability by first thinking about the likelihood of events and associating this with the language of probability.
Present students with a list of events, from the everyday to the rare, and ask them to decide which words best describes the likelihood of the event occurring.
The previous activity introduced students to the language of probability and began the process of understanding that probability exists on a scale and as such has an associated value. To further impress this concept upon the students you should now ask them to now place the events on the scale shown below.
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| Impossible | Highly unlikely | Unlikely | Even chance | Likely | Highly Likely | Certain |
To begin to link these concepts to more rigorous mathematics you should now introduce the students to the mathematical values associated with probability.
Probability (Impossible Event) = 0 [shorthand P(Impossible) = 0]
Probability (Certain Event) = 1 [shorthand P(Certain) = 1]
Ask students to make estimates or determine number values for the probabilities of the other words associated with the likelihood of events.
The Comment activity enables them to practise using the language.
Learning Intention: To estimate or calculate the probability of an event occurring.
Consider the list of simple experiments or events shown below and list all the possible outcomes for each event:
Each student should think about the following topic: How could you estimate or calculate the probability of each of an event occurring? Once you have a possible solution, find a partner and share your ideas in a Think-Pair-Share activity.
Comment: Share one idea from your discussion. Then comment on other students' possible solutions, expanding on their thinking wherever possible.
Sample Spaces
Once students have gotten a handle on the basic language of probability and the range of values, the next step is learning to calculate how likely an event is, is to consider all possible outcomes, i.e. also consider the options that could occur if your desired event doesn’t. To do this the students will be presented with a series of basic experiments that have definite outcomes that they can record.
Define this list of all possible outcomes as the sample space. Make sure that the students understand that the sample space itself is the list - not the size of the list (or the number of options).
Once the students have listed the possible outcomes, they should Think-Pair-Share the following topic: How could you estimate or calculate the probability of each of event occurring?
If the students are struggling, direct them to think about the numerical range of probabilities possible, what an ‘even chance’ means (i.e. consider the simple example of a coin toss) and how the list of possible outcomes might help in harder examples.
The discussion is a scaffold for writing the comments. The comment ensures every student is accountable. It also becomes a record of the students' thinking and a reference point for students when they add ideas to their Video Projects.
Learning Intention: To understand the equations used to calculate the probability of an event.
Look at the following equation:
If you are writing down the probability of an event you shouldn’t just write the number value but also what it represents. For example, if you want to write the probability of tossing a head you could write any of the following:
Probability (Tossing a Head) or P(Tossing a Head) or P (Head) or P(H).
Copy the examples the teacher has explained and written on the board into your workbook.
Work on your own to complete the questions listed by the teacher in your workbook.
Work on your own to complete the next set of probability problems provided by your teacher. This time you must express your answers for each question as a fraction, a decimal and a percent.
Comment: What are the advantages and disadvantages of expressing probabilities in a variety of ways (fractions, decimals or percentages)? Comment on other students' comments that you agree or disagree with. Explain why.
Calculating Probabilities
[You will need: Section 6.2 of the Probability Workbook as handouts]
At this stage you should formally introduce the equation for probability. Provide the class with the definition:
Note that sometimes this is written using the shorthand notation P(E) or that the E can be replaced by the specific type of event. For example if you want to write the probability of tossing a head you could write any of the following: Probability (Tossing a Head) or P(Tossing a Head) or P (Head) or P(H)
Go through a number of examples of problems on the board and have the students copy them into their workbooks.
Select several simple questions from Section 6.2 (p.129) of the Probability Workbook for the students to try.
Explain to the students that probabilities may be expressed in a variety of ways (fractions, decimals or percentages). Remind them how to convert between these different types of numbers and then have them finish several more probability problems from Section 6.2 (p.129) of the Probability workbook where they should present answers in all 3 forms.
Learning Intention: To understand the odds of winning a lottery.
Brainstorm: What you already know about Lotto?
After watching the You Tube video on the Probability of winning the Texas Lotto and reading Info About the Odds from the NSW Office of Liquor, Gaming and Racing website, work with a partner to answer the following questions to the best of your ability.
Scene 2: You decide…
Comment: Share some ideas from your discussion. Then work on your scripts for Scene 2 in Creator. Comment on other students' suggestions and build on any ideas that might be useful for them, yourself, or other groups.
Looking at Games of Chance
Re the video on winning the lottery in Texas, reassure the students that they don’t need to understand the maths, but just try to get a handle on the process and the comparison made at the end about being struck by lightning.
Students are asked to brainstorm what they already know about lotto as a whole group on the whiteboard.
Prompt Questions:
How many numbers to choose from?
How many numbers are drawn?
Is there replacement?
To complete the questions, students will also need to know how to convert numbers into scientific notation. As they haven’t done this before, you will need to go through the process with them.
Scientific Notation:
- Where Mantissa is a number between 1 and 10
- Index is the number of places the decimal place moves
Step 1: Find the starting place of the decimal point in your original number
Step 2: Move the decimal point so you have a number between 1 and 10
Step 3: Count the number of places the decimal point has moved
Step 4: Write the number as
Ongoing Assessment: Stage 2
In class or for homework, have the students discuss and complete the work on their second scene where they will explain what they have learnt since completing the last section.
This section could include a discussion of the range of values of probability, how you can calculate probability and sample space.
Learning Intention: To understand complementary events.
Watch the video: Complementary Events
What are the complementary events for?
What is the complementary event to:
Tossing a die and getting heads
Winning the lotto
Tomorrow is my birthday
Answer the questions to Exercise 15E on Complementary Events in your Mathematics Exercise Books. Please be certain to label the questions carefully.
Comment: Share at least one fact you learnt about complementary events.
Complementary Events
[You will need: Complementary Events Sheets – page 1, page 2, page 3]
Complementary events are multiple events for which the probabilities have a total of 1. In other words, complementary events cover all possible outcomes to the probability experiment.
The sum of the probability of an event and its complements equals 1.
To calculate the probability of an event, subtract the probability of its complementary events from 1.
Discuss the complement of everyday events,
Select questions from the Complementary Events sheets for students to answer.
Learning Intention: To gather experimental data on dice probability
Theoretical Data is the data you expect to get from an experiment based on your knowledge of probability.
Experimental Data is the data you actually gain from completing an experiment.
Over time, experimental data should give similar results to theoretical data after many attempts at an experiment.
Work with a partner to complete the activities on Dice Probability.
Comment: What are some observations you made during the experiment. Comment on the observations of other students, noting similarities and differences in your outcomes.
Experimental vs Theoretical Probability
One pair of six-sided dice is needed per group and a calculator in order to work out the probabilities. Also, lined paper should be provided to each group in order for them to complete a report of the investigation.
Learning Intention: To predict what coins will be drawn out of a bag.
The teacher will hold a sack at the front of the room. The sack is filled with an assortment of coins. In groups you will try to determine how much money is in the sack. To solve this problem you will need to know:
Comment: Add your group's guess to the Comment box. The first group to predict correctly wins. Discuss how you will explain your prediction.
Making Predictions
[You will need: An opaque sock or sack filled with an assortment of money – make sure to have multiples of some coins and none of others]
In this activity students will work in groups to attempt to determine how much money is in the sack.
Give each group 2 minutes (keep it short so they have to keep thinking while they play) to plan how they want to play.
Learning Intention: To understand the function and importance of probability.
Work with a partner. Student A asks: Do we need probability? Student B responds.
Then Student A repeats the question "WHY?" 4 more times. Student B responds, taking time to think and probe the question in more depth.
Then reverse roles and Student B asks the question: Where do we need probability. Student A responds.
Then Student B repeats the question "WHY?" 4 more times. Student A responds, taking time to think and probe the question in more depth.
Ongoing Assessment: Scene 3 - add ideas to The Structure Tool in Creator where you explain what you have learnt since completing the last section. This section could include a discussion of complementary events and a comparison of theoretical and experimental probability.
Comment: Discuss what happened in the "Five Whys" strategy. Which response - the 1st, 2nd, 3rd etc gave the most thoughtful answer? Explain why. Comment on other students' comments, noting similarities and differences in your responses and experiences of using the "Five Whys" strategy.
The 5 Whys of Probability
This activity forces the students to think deeply about the responses rather than just providing superficial answers (even if they do this for their first response).
Ongoing Assessment: Scene 3
In class or for homework have the students discuss and complete the work on their third scene where they will explain what they have learnt since completing the last scene.
This section could include a discussion of complementary events and a comparison of theoretical and experimental probability.
Learning Intention: To understand how to represent probability on a tree diagram.
Watch the Khan Video: Count Outcomes Using Tree Diagram.
Conduct an experiment on probabilty and document it on a tree diagram.
Comment: How does a tree diagram help you to determine probability? Comment on other students' comments, building on their ideas.
Creating Sample Spaces for Compound Events
Watch video on tree diagrams
A Tree Diagram is necessary in any example where there is more than one stage to the probability experiment.
The tree diagram must branch out once for every stage of the probability experiment.
Once the tree is drawn, the sample space is found by following the branches to each end.
Learning Intention: To understand the probability of compound (or multiple) events.
Follow the teachers explanation of how to calculate the probability of compound (or multiple) events. For example, the probability of getting 2 heads when you toss 2 coins.
Copy notes on the process and the specific examples covered. Complete the selected exercises on compound events of the Probability Workbook. Remember to show all working, including how you determined the sample space, for each question.
Comment: What might happen if each outcome of an event is not equally likely? For example, if you roll a weighted die and the chance of rolling a 3 is higher than the other 5 numbers. How could you calculate the probability in this case? How could you create the sample space?
Copy the example shown by the teacher into your workbooks and then attempt the selected questions from p.137 of Section 6.3 of the Probability Workbook.
Calculations involving Compound Events
See Probabilities of Compound Events (Khan).
Go through the process of calculating the probability of compound events by showing several examples for students.
Highlight the steps required to make such calculations:
The students should complete selected exercises.
Extension:
Look at probabilities where each outcome is not equally likely.
Ask the class to consider what would happen if each outcome was not equally likely, for example, if you used a weighted die or coin. Show an example of a tree diagram where the probability (or weighting) of each single event was shown (Example 3 on p.135 in Section 6.3 of the Probabilty Workshop is a good option). Demonstrate that to calculate the probability of compound events in this case you multiple the probability of each single event in the sequence. Ask the students to complete selected questions from p.137 of Section 6.3 of the workbook.
Learning Intention: To understand dependent and independent events in probability.
Watch the teacher demonstration of a “colour lotto” as they draw objects from a container. Consider what will happen as the objects are drawn from the container and not replaced. Does the probability of drawing a particular colour go up or down? Does this probability depend on what was drawn previously?
You are the winner in a quiz show and can choose a prize from behind 3 locked doors. Behind 1 door is a new car. Behind the other 2 doors are goats. When you have made your choice the host opens one of the other doors to reveal a goat. Should you stick with your choice or switch to the other one? Or does it make no difference? Watch Let's Make a Deal.
On your own, read the problem and think about what you would do in this situation. When the teacher asks you to, share your answer and the reason why with a partner. You can even try it at Monty Knows.
Participate in a class survey of results. What would this class choose to do in that situation?
The watch Probability and the Monty Hall Problem (Khan Video). You can also practise the concept.
Scene 4: You decide…
Discuss with your group what you will talk about in Scene 4 of your Video. Add ideas to Scene 4 in the Structure Tool.
Comment: What did you learn about probability in this Update? Comment on other students' comments, building on their ideas. Add links to other websites/information that add more information or other problems relating to conditional probability.
[You will need: A clear container with a selection of coloured object]
Look at the case where compound events are related (for example the. non-replacement of balls during the lotto draw). Show the class a clear container with a number of different coloured balls, counters or dice. Ask them what happens to the probability of drawing a particular colour (say red) as you draw objects from the container. Will it increase or decrease? Does it depend on the colour of the previous ball drawn?
Extension (For the Level 1’s and maybe 2’s if they are coping okay with the concepts):
Present the class with the Monty Hall problem. Give them a few minutes to discuss (Think-Pair-Share) and consider what they would do and then take a class survey of who would stick and who would change. Go through the theoretical explanation of why you should always switch (refer to the notes on websites such as Monty Hall Problem for more information)
If there is time you could run the game several times to begin to establish and experimental probability and/or have each member of the class set up their own version that they can run with a partner.
Draw three doors or boxes on the board or in student’s workbook. On a piece of paper right down the door that has the major prize. Select a member of the class to play with and then run through the problem. Record if they won and whether or not they stayed or switched on a table like the one shown below:
| Won | Loss | |
| Switched | ||
| Stayed |
Ongoing Assessment: Scene 4
[You will need: Video workbooks]
In class or for homework have the students discuss and complete the work on their fourth scene where they will explain what they have learnt since completing the last section of the workbook.
This section could include a discussion of compound events and a comparison of independent and dependent events.
Learning Intention: To reflect on what you have learnt about probability.
Repeat the brainstorming activity you completed at the start of this Learning Module.
Repeat the game "Pig".
Comment: Are the results of the game different from when we first played? What are some things you have learnt in the Learning Module? Comment on the comments of other students, building on their ideas where possible.
Learning Survey
[You will need: Small post-it notes]
Repeat the original brainstorm activity (Probability Mind Map) and see how the class’ responses have changed.
Repeat the game Pig and ask students to compare the number of times they remained standing for each round the first time they played at the start of the unit compared with this second game.
Learning Intention:To analyse the effects of relying on probability.
Watch an episode of The Simpsons (Season 5) on Marge’s gambling addiction.
Then Read a Fact Sheet on the effect of gambling on real people and society: The Facts: Gambling in Australia.
Note that there are three parts to the Comment section.
Comment: Do you think that using humour about a serious issue like Gambling is effective? Why/Why not? Comment on other students' opinions, giving reasons for why you agree or disagree with them. Then comment on what do you think are the most serious negative effects of gambling. Find other information about the effects of gambling, describe at least one negative effect, and post a link to the information in the Comment box.
Social Implications of Probability
Background on Marge's gambling addiction - Season 5 of the Simpsons. You will have to purchase this episode.
Students then become active knowledge makers, finding, contributing, and commenting on information they find. Add comments, ask questions and comment on the students' comments, modelling how to give reasons and evidence for their opinions.
Learning Intention: To research how probability affects society.
Your group will be assigned one of the following topics for Scene 5 of your video:
Research your assigned topic. You may use books, the internet or your own experiences and opinions to form the basis for your discussion of this topic.
Discuss with your group what you will talk about in Scene 5 of your video and then add notes to your project in Creator in Scholar.
Comment: What is one important issue you will include in Scene 5. Comment on other students' comments, particularly if you can add more information that will help other groups on their assigned topic.
Ongoing Assessment: Stage 5
Assign each group a different discussion question related to probability in real life (refer to questions in Year 9 Intermediate Signpost textbook p 184, Pearson).
In class have the students discuss and complete the work on their last scene where they will discuss the social implications of probability.
If the groups are struggling to find useful information try directing them to the following websites:
General info on the use of probability in society
Resources for Teaching and Learning about Probability - ERIC digest
Collision Insurance, Automobile Accidents, and Trees: Making a Decision with Impact
Medical Screening:
Screening or Not Screening for Lung Cancer: Consider the Benefit
Screening for Dementia red flags in primary care: Is it time?
Repair Warranties:
Mentioned on Wikipedia under ‘Probability’ and ‘Applications’.
Risk Worksheet: Examples of Expected Value Calculations
Learning Intention: To give feedback on other students’ works and then revise my own.
Check your Notifications for Feedback Requests: You have received a Feedback Request. Click on this link to take you to the work you have been assigned to review.
Go to Feedback => Reviews => Review Work. Rate the work on each criterion and explain why you gave the work that rating. Make in-text comments at Feedback => Annotations.
Submit your feedback once it is finished at About This Work => Project => Status. You will not be able to submit your review until you have completed the Review and Annotations.
For more information, see Reviewing a Work and Submitting a Review and Annotations.
Revision Phase
The next stage of the writing process is to revise your own work.
Check your Notifications for a Revision Request: You have received a Revision Request. Click on this link to take you to the most recent version of your work. Then go to Feedback => Reviews => Results to see the reviews and Feedback => Annotations to see in-text comments. Once you have incorporated all of the feedback (Reviews/Annotations) from your peers, click “Submit Revision” below the work.
You can also write a self-review, explaining how you have taken on board the feedback you received.
For more information, see The Revision Phase.
Comment: Do you have any questions about Scholar? Make a comment in this update. If you think you have an answer to another student's question, please answer it. Start with @Name.
This Update covers two phases of the writing process in Scholar - Review and Revision. Post it directly into a Community when students are up to this phase of their writing projects. They will also receive Notifications.
Refer to Analytics to monitor how students are progressing with their writing and their reviews.
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