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.

### 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)

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 pre-cut 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.

• Students may believe that the probability of an event occurring in a two or three-stage probability experiment is the same as the probability of the event occurring in a one-stage 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.

Key language: compound event with and without replacement, simulation, two-stage compound event, relative frequency, ‘and’ and ‘or’ statements, ‘inclusive or’ and ‘exclusive or’ statements.

## What you need:

• Lesson plan (Word)

• Teacher's notes (Word)

• Egg-cellent worksheet (Word)

• Egg roulette (Video)

• Egg roulette simulation (Excel, macros enabled)