Louis' ice cream business
Problem
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?
You can model Louis' options using yellow cubes for sunny days and blue cubes for rainy days.
Put two blue and two yellow cubes in a bag so that the colour of the cubes cannot be seen.
Now decide where Louis should put his first icecream stall. The table tells you how many dollars he can expect to make per day in each location:
| Rain (blue) | Sunny (yellow) | |
|---|---|---|
| Bandstand, A | 88 | 168 |
| Beach, B | 50 | 256 |
| Beach cafe, C | 100 | 100 |
Now pick a cube How much did Louis make?
- Was this a wise choice?
- If you were to choose again, would you stick with the same option or change to another?
- If you would not change, why not? If you would change, which would you choose and why?
- Which do you think would be Louis' best option over a month (30 days)?
See if experiment confirms your choice.
If you work in a group of three, each of you could choose one location and see whose location does best.
You could record the weather each day, then compare the financial outcome for each location day by day, and see which did best over the month.
Which location did best? Does this surprise you?
Since rainy and sunny days are equally likely in this model, which location would be Louis' best option over a 6 month season?
Now think about the model we have set up: what assumptions have we made?
Louis opens a second icecream stall!
Business is doing quite well, so Louis decides to invest in another icecream stall further round the beach, where it won't compete with his first stall, and where there are likely to be more people.
There are three locations he thinks might do: on the beach (J), under an umbrella in front of the pier (K), or in a cafe at the end of the pier (L). He will have to be careful which location he chooses, because some of them will mean he loses money if it rains - although they will also be good earners if the sun shines. Also, Big Frankie is in charge here, and he has to be paid for letting Louis open a business on his patch - and Big Frankie will expect to be paid whatever the weather.
The table gives Louis' expected income and outgoings per day:
| Payment to Big Frankie |
Rain (blue) |
Sunny (yellow) |
|
|---|---|---|---|
| Beach (J) | -30 | -300 | 700 |
| Umbrella in front of the pier (K) | -50 | -100 | 300 |
| Cafe at the end of the pier (L) | -200 | 50 | 500 |
Which location do you think Louis should go for?
What is his expected income in each location over a period of a month (30 days) if he considers just the income on the icecreams?
What is his expected income in each location of a period of a month when he takes Big Frankie's payment into account?
So which location would be best for him?
Is it worth his while opening this second stall?
Would your decision be different if Louis' business will fold if his total income drops below zero?
Do a running total for a 30 day experiment and see if any of the possible locations mean Louis goes out of business.
You may find this spreadsheet helps with this.
You will need to decide what you're going to do if the first day or two are rainy - does Louis go bust immediately, or do you think he has a cushion from his first stall?
Developing the model
This isn't very realistic, however, so now we are going to make the model a little more sophisticated.
Louis lives in an area where rainy and sunny days are both as likely as each other. But he notices that if it is sunny one day, it is more often sunny the following day, and if it is rainy one day, the next day is more likely to be wet.
For the first day, put two yellow and two blue cubes in the bag. Take one out and note what colour it is, then replace it in the bag.
- If it is yellow (so the first day was a sunny day), add another four yellow cubes to your bag.
- If the first cube is blue (so the first day was a rainy day), add another four blue cubes to your bag.
For day 3 and successive days, replace the cube for day 2, and make sure that there are the appropriate cubes in the bag given the weather on the previous day.
Use the cubes to give a weather pattern over 30 days. How many sunny/rainy days were there?
Compare your results with other groups? If you average the data across the class, what do you get?
Can you predict what the expected number of sunny days might be over 30 days?
To check what we would expect, draw a tree diagram for days 1 and 2, using the probability of a sunny or rainy day as given by the cubes.
What is the probability of a sunny day on day 2? Does the answer surprise you?
Now extend your tree diagram to a third day.
What is the probability of a sunny day on day 3? Again, are you surprised?
What does this suggest for the probability of a sunny day on day n? Can you justify your answer?
Can you think of ways you could improve this model further?
Teachers' Resources
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.