Egg roulette: Part 2

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 Egg-cellent 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 Egg-cellent 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).

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 Egg-cellent worksheet.

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 multi-stage 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 pop-up 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 on-screen 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.

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 two-step 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.