### Prize Giving

What's the fairest way to choose 2 from 8 potential prize winners? How likely are you to be chosen?

# Louis' Ice Cream Business

##### Stage: 4 Challenge Level:
Louis' Icecream Business

Louis plans to set up a new icecream business by the seaside.  But where should he set up his first stall?

If he sets up shop at the Beach Cafe (C), the weather won't matter.  On the other hand, if he puts his stall on the beach (B), he'll do much better if it's sunny, but much worse if it rains.
He could go for the bandstand (A) instead ...

What should he do?

### Why do this problem?

This problem provides a context for probability and risk as an aspect of decision-making in the real world.

The final section also provides a context for using a probability tree and sampling with replacement to consider the issue of dependence and independence.

#### Possible approach

Put students into groups of 3, and have each group collect the equipment they need:

• 2 yellow and 2 blue cubes
• an opaque bag to put them in

Take the students through the scenario for Louis' first icecream stall, ensuring everyone understands how the model works, and how they are to collect data.  Groups of 3 are ideal, because then each person can take one location and see how it performs.

Students will need to consider what data they need to record, and how best to record it.  Once they have their data, they will need to find a way to analyse it - a 2-way table works well here, with the three locations along one axis and the two alternatives of income on a sunny day and income on a rainy day on the other.

#### Key questions

Given that we don't want to have to use experiment to find out which is Louis' best option over 6 months, how can we work out what his expected daily income is in each location?

What are the assumptions in the model?
How realistic are they?
Are they justified in a first attempt at the model?

#### Possible extension

Move onto Louis' second icecream stall and the development of the model (see below).

#### Possible support

All students should be able to carry out the experiment, once they have understood the scenario and done one or more initial trials all together.

Students who find it difficult to move from the results of their experiment to the expected results could be helped to structure their thinking by using a table of their results.
Alternatively they could complete a 2-way table for what would happen if there were 15 sunny days and 15 rainy days (which is what we would expect the cubes to give us on average).

### Louis' second icecream stall

#### Possible approach

Encourage students to use the new values for sunny and rainy days first to find out what the expected income is for each location.
Then factor in the payments to Big Frankie, and see how that changes the results.

How does it change things if a run of rainy days could mean Louis goes out of business?
Suggest students keep a running total of his income during their 30 day experiment - this spreadsheet could be used to help with this.
Students will need to decide what to do if the month starts with rainy days - does he go bust immediately, or will they assume a cushion from his previous stall?

#### Key questions

What difference does it make when we factor in the payments to Big Frankie?

Is it worth Louis' while opening this stall?
This is not a question with a single right answer, but will depend on students' perception of risk.

#### Possible extension

The model is developed by creating a more realistic scenario for the weather, which students can explore first by experiment then by analysis of a probability tree.

The (initially) surprising result is that despite this change, the probabilty of a sunny day at any stage is still 1/2!

This provides a context to explore what we mean by independent and dependent events.  It appears that the weather on day 2 is dependent on the weather on day 1 - is this the case, or not? Clearly the weather other than on day 1 is dependent on the weather the previous day, but on average there are as many sunny days as rainy days.
This is an example of a Markov chain.