Probability is the mechanism underlying statistics, and the basis for the types of statistical tests that might be familiar to A-Level mathematicians. Although they are often nominally grouped together in most school curricula, this strong link between probability and statistics is often overlooked.
Imagine an ornate, old-fashioned pocket watch: you might only ever look at the face to see what time it is, but take the time to open it up and you’ll see the mechanism ticking away relentlessly beneath. No matter how much the time we read off the dial seems to elude or surprise us, the watch (unless broken) works in the same way – in a complex but predictable pattern. This metaphor only goes so far – especially when we consider outputting a range of possible values – but it is worth considering why the mathematics curriculum has become siloed in this way, and whether it is helpful for later mathematics.
While statistical inference can be a dark art with many a bear trap waiting to snare the unwary, it is possible to begin to explore the underlying ideas with secondary students in a straightforward way that both supports their understanding of probability in the short term and lays the groundwork for the concepts needed in further study.
Encouraging students to explore “urn models” can lead to important insights into the nature of random chance in the study of statistics. An urn model is, at its simplest, a bag containing coloured marbles. Two urn models can be considered isomorphic (that is, completely identical) if the proportions of coloured marbles within each are the same. Furthermore, the urn can be considered isomorphic to seemingly unrelated scenarios if the proportions of the coloured balls in the pot represent the probabilities in the scenario. In this way, an urn model or series of linked urn models can be used to simulate increasingly complicated events such as making predictions of the weather by converting forecasts into proportions of coloured marbles.
The attached resource is a simulation of an urn model built in an excel spreadsheet. It replicates the classic probability experiment of drawing coloured marbles from a pot, but allows learners to approach it from an inferential standpoint - namely, if you can only draw one ball at a time, and then replace it, can you reliably work out the contents of the pot?
There are some really deep conceptual ideas tied up in what is a very simple experiment:
- The isomorphism of probability spaces means that a physical experiment can be simulated and the outcomes will be equivalent,
- repeated sampling will give different outcomes each time,
- the more trials that take place, the more stable the picture that develops,
- experimental parameters affect the overall confidence in any inference,
- in the long run, the frequency will reflect the proportions of coloured marbles in the pot.
While a physical demonstration gives students a much clearer understanding of the rules of the experiment and is a recommended starting point, the flexibility of the simulation means that many different parameters and the relationship between them can be explored.
Can you make a prediction about the contents of the pot after 1 trial? How about 2? Or 3?
How many different coloured marbles are in the pot?
What’s the minimum number of trials needed to be confident in the set of colours if there are 5 marbles in the pot? Is this the same if there are 10 marbles?
If graphing, after how many trials does the graph appear to stabilise? Is this the same for different sized pots?
Do certain numbers of trials make it easier to predict the contents of the pot?
What proportion of the pot is made up of each colour?
If we did the experiment 10 times and got 5 green, 2 yellow and 3 blue, how confident could you be that they came from the current pot?
If we were able to conduct an infinite number of trials, what might the graph look like?
If we only were only allowed 5 trials before resetting the graph, but could repeat this process as many times as we like, how could we decide what is in the pot?
As you can see there are endless avenues of enquiry that can be explored which prod at the very beginnings of the concepts needed at a higher level, but also begin to develop a deeper understanding of proportion as a model that reflects the real world, leading to the theoretical probability calculations that more commonly form the starting point of classroom exploration in this area.