Egg roulette: Part 2
Year level: 9
Strand: Probability
Lesson length: 60 mins
In Part 1, students were introduced to probability without replacement, by exploring the game of egg roulette – 12 eggs, 3 raw and 9 cooked. Contestants take it in turns to choose an egg and smash it on their head hoping to avoid the 3 raw eggs! In this lesson, students consolidate and make deeper connections using simulation to address misconceptions on this abstract concept. Note that the same resources are also available in Part 1 as Part 2.
Curriculum information
Achievement standard
By the end of Year 9, students determine sets of outcomes for compound events and represent these in various ways, assign probabilities to the outcomes of compound events and design and conduct experiments or simulations for combined events using digital tools.
Content description
 Students list all outcomes for compound events both with and without replacement, using lists, tree diagrams, tables or arrays; assign probabilities to outcomes. AC9M9P01
 Students calculate relative frequencies from given or collected data to estimate probabilities of events involving ‘and’, inclusive ‘or’ and exclusive ‘or’. AC9M9P02
 Students design and conduct repeated chance experiments and simulations, using digital tools to compare probabilities of simple events to related compound events, and describe results. AC9M9P03
General capabilities
Numeracy progression
 Probabilistic reasoning (Level 6)
Digital literacy
Critical and creative thinking:
 Draw conclusions and provide reasons (Level 6)
Assessment
Exit ticket
The exit ticket is included in the Teacher’s notes document. Note that there are three exit tickets per A4 landscape page to precut before distributing.
A box of chocolates contains four chocolates that look identical but three are caramel (yum) and one is a peanut (yuk).
Use your knowledge of probability, with a diagram, to help me decide whether I should eat one or two chocolates.
Some possible answers (in order of increasing sophistication) include the following.
 If you are anaphylactic you should have no chocolates. If you have only 1 there is still a 14 chance you get a peanut and could have an allergic reaction! It’s too big a chance.
 If you only have one chocolate there is a 14 chance you get a peanut and
34 caramel, so it seems like quite a good chance you get a chocolate you will like, so worth taking.  The table shows the outcomes if you have two chocolates. Of the 12 outcomes, in 6 of them get two caramels (CC); that is, there is a 14 chance you get two caramels but also a 50% chance you get a caramel and a peanut, so you may not want to do this if you really hate peanuts. Just having one chocolate might be a better choice as there is just a chance you get the peanut.

C1 
C2 
C3 
P 
C1 

CC 
CC 
CP 
C2 
CC 

CC 
CP 
C3 
CC 
CC 

CP 
P 
PC 
PC 
PC 

On your first pick you have a 14 chance you get a peanut and 34 caramel, so it is worth taking the risk. If you get a peanut, although that is not nice, you know all the rest are caramels so you should keep going and eat them all! If however your first pick is a caramel, you have a 13 chance of getting a peanut on the next pick as there are only 3 chocolates left, so it may be worth stopping.
Areas of challenge
 Students may believe that the probability of an event occurring in a two or threestage probability experiment is the same as the probability of the event occurring in a onestage experiment. The impact of an experiment without replacement is Illustrated via egg roulette, where it is clear that eggs are ‘used up’ and no longer appear in the sample space in later stages. This misconception is addressed through detailed modelling of the sample space using a table approach for the first two stages.
 Students may assume that outcomes are equally likely. The distribution of cooked / raw eggs has been deliberately chosen to be heavily skewed towards cooked eggs. Students can be asked questions such as, ‘Are there more cooked or raw eggs?’ ‘Is it more likely [person] gets a cooked or raw egg? Explain why.’
 Students may be confused with the discrepancies between theoretical and experimental probabilities. An example such as tossing a coin once could be used where it is impossible to get half a head. Using the simulation to increase to a large number of trials allows for illustration of the Law of Large Numbers, reducing the discrepancy between the two.
Prerequisite student knowledge and language
Key language: compound event with and without replacement, simulation, twostage compound event, relative frequency, ‘and’ and ‘or’ statements, ‘inclusive or’ and ‘exclusive or’ statements.
Learning goals
Learning intention
 We are learning to understand probability without replacement.
 We are learning to analyse compound events visually and numerically.
 We are learning to understand the changing probabilities at each stage, including by simulation.
Success criteria
By the end of the lesson, students can:
 correctly identify how probabilities change at different stages in a without replacement probability experiment
 use an array to find probabilities for twostage compound events
 conduct repeated chance simulations
 calculate relative frequencies from collected data to estimate probabilities involving ‘and’ and ‘or’ statements.
Why are we learning about this?
Understanding probability is foundational in mathematics, aiding progression to advanced statistics topics. It intertwines with realworld scenarios, enriching our decisionmaking and analytical skills. Examples include in genetics, where trait inheritance follows probability patterns, and in card games where the odds may change with every card drawn. Understanding changing probabilities helps us comprehend the world and make informed decisions.
Learning hook 30 mins
This lesson follows from the previous lesson Egg roulette: Part 1. Access the video here that was introduced in Part 1 if a review is required or refer to the online version of the lesson where the video is embedded in the learning hook.
Refer students to the Eggcellent worksheet and pair students together. If you had not reached the Egg roulette: exploring the probabilities activity last lesson, students can continue to complete the questions on the worksheet. Otherwise come together as a class and use the following questions to ensure a deep connection with the content has been made and to provide further opportunity to clear up misconceptions.
Refer to the completed table in the Eggcellent worksheet and ask the class to consider the scenario before the game started. Ask the following questions.
What is the probability that both Denise and Miriam choose raw eggs?
 6132 ≈ 0.045 ≈ 4.5% very unlikely.
What is the probability that Miriam chooses a raw egg on her first pick?
 312 = 14 = 0.25 = 25% or using the table counting all the cells starting with R
as 33132 = 14 .
What is the probability that Denise chooses a raw egg on her first pick? What do you notice? Why?
 Counting all the cells ending in R gives 33, therefore 33132 = 14 . The initial probability that Miriam and Denise both choose a raw egg on their first pick is the same, as the game is fair. Initially both Miriam and Denise are equally likely to get the same number of raw / cooked eggs (though this changes as the game progresses).
What is the probability that the first egg is raw (Miriam) and the second egg is cooked (Denise)?
 Counting all the cells with RC gives 27, therefore 27132 = 944 ≈ 0.204 ≈ 20.4%.
What is the probability that exactly one teacher chooses a raw egg on their first pick?
 Counting all the cells with RC plus CR gives 27 + 27 = 54, therefore
54132 = 922 ≈ 0.409 ≈ 40.9%. ‘Exactly one’ is an ‘exclusive or’ phrase and does not include the possibility that they both picked raw eggs. This question could be rephrased as ‘Miriam or Denise, but not both, chooses a raw egg on her first pick’.
What is the probability that at least one of the teachers chooses a raw egg on their first pick?
 Counting all the cells with RC plus CR plus RR gives 27 + 27 + 6 = 60, therefore 60132 = 511 ≈ 0.454 ≈ 45.4%. At least one is an ‘inclusive or’ phrase and does include the possibility that they both picked raw eggs. This question could be rephrased as ‘Miriam or Denise or both, choose a raw egg on her first pick’.
It is important to distinguish between the language of OR (one of the two) versus AND (both of the two).
Explore 20 mins
Draw up the table below on the whiteboard showing the theoretical probabilities of the 4 different combinations that you have just covered and also from the activity completed in the Eggcellent worksheet.
Outcome 
Theoretical probability 
Cooked – Cooked 
72132 ≈ 54.5% 
Cooked – Raw 
27132 ≈ 0.204 ≈ 20.4% 
Raw – Cooked 
27132 ≈ 0.204 ≈ 20.4% 
Raw – Raw 
6132 ≈ 0.045 ≈ 4.5% 
Differentiation (support): amend the activity to include fewer eggs: for example, 6 eggs, with 2 raw and 4 cooked.
Differentiation (extension): various options are given below.
 Introduce a probability tree diagram to model the different outcomes for the first two picks.
 Explore the product rule for multistage probability events.
 Ask students if there were 6 raw and 6 cooked eggs would each of the outcomes above be equally likely? The perhaps surprising answer is no, as Cooked–Cooked and Raw–Raw each have a 612 × 511 = 30132 ≈ 22.7% chance of occurring, whereas Cooked–Raw and Raw–Cooked have a higher chance of occurring, calculated as
612 × 611 = 36132 ≈ 27.3%.
Egg roulette – simulation
(Note that the Excel spreadsheet requires you to enable macros. Download the Excel spreadsheet from the What you need section. A window may popup asking whether to make file a trusted document. Select Yes, if not you can change setting in the Trust Center, located in File>Options>Trust Center>Trust Centre Settings.)
In pairs, students conduct probability simulations using 50 trials to gain data on:
 the likelihood of each of the outcomes above for the result of the first two picks and
 the number of raw eggs they would expect to get in a full game
Refer to the activity and Teacher’s notes. Distribute the Egg roulette simulation Excel spreadsheet to students and demonstrate onscreen how to run the simulation and answer any questions about the questions on the sheet. Give students time to complete the activities, monitoring their progress and offering guidance using the Teacher’s notes.
Differentiation (extension): introduce expected value and ask students how many times they would expect to get 0, 1, 2 or 3 eggs in 100 trials or 500 trials.
Differentiation (support):amend the activity to include fewer eggs, for example, 6 eggs, with 2 raw and 4 cooked.
Summary and reflection 20 mins
Conduct a class discussion reviewing the results of the simulation and the overall learnings from the lesson and reserving time for students to complete the exit ticket described in the Assessment section.
Key points are as follows:
 In experiments without replacement, probabilities change at each stage and both the denominator and sometimes numerators need to change (depending on the previous pick and which outcome you are looking at).
 Tables / arrays are a good way to help visualise twostep probability experiments (like the first two picks).
 The game is initially fair, but as it progresses probabilities change owing to the without replacement property.
 Owing to chance, experimental probabilities tend not to match theoretical probabilities.
 The law of large numbers: the larger the number of trials, the closer experimental probabilities are to theoretical probabilities, which could be demonstrated if time permits by combining simulation data to give a class result.
 The Gambler’s Fallacy: random events are not impacted by past events.