Experimental Probability

I love teaching probability. For me, it is not necessarily about working out the likelihood of an event occurring but working out the event when given the probability.

What I mean by this is it’s not difficult to understand the probability of a fair dice landing on a 3 is 1/6 or the chances of picking a King at random from a fair deck of cards is 1/13 . Probability, as seen in this way is normally taught at 11 years old often with little confusion or misconception. So when I teach it further on in school life I like to reverse and deepen the process through a project that spans a sequence of lessons.

Introduction
I present the slide below to the class and ask them to work in small groups to consider what numbers may be on the spinner to fulfil the probabilities. This way we recap basic number probabilities as well as theoretical probabilities.

The interesting thing here is when the class realise the probabilities overlap. For instance which number is a multiple of 3 as well as odd?

Development Phase

To build on the introduction I set out the learning objectives and success criteria for the lesson as shown.

The task is to create and test a hexagonal spinner given a fixed set of probabilities and consider how the actual results may be different to what could have been expected.

Choosing the numbers
Depending on the ability of the class I choose between 5 or 8 different overlapping events. For instance, for the least able I would include 5 events with basic number properties such as factors, multiples, odd and evens but for most able we would also consider triangular and cube numbers as well as inequalities. When I taught this to a level 6 year 9 class I used the slide shown below.


A possible set of numbers in this case could be 0, 2, 3, 8, 10 and 15.

Constructing the spinner
Step 1. Using a piece of card and pair of compasses students create a circle of fixed radius. Using this fixed radius they mark of points along the circumference.

Step 2. Join the arcs to the centre and create chords between them.

Step 3. Cut out the hexagon and pierce a pencil through the centre.


The illusion is that they create a regular hexagon. However, because Circumference = pi*D and pi is approximately 3.142 they have created five equal segments and one slightly smaller. This helps to introduce bias.

Experiment

Once the numbers have been calculated and the spinner created students work together to decide on a suitable sample size for the experiment. Here we talk briefly about relative frequency. Typically, most groups choose 30 spins since it is a multiple of 6 and doesn’t take too long. Before they begin the experiment we discuss, as a class, the expected frequencies of each category as shown.


Over the course of the 30 or so spins the spinner can become slightly damaged with the pencil slipping in the hole through the centre. This helps to increase bias in the experiment making the actual frequencies different from the expected and serves as an excellent discussion point in the plenary.

Who has the best spinner?
To wrap up the first lesson of the two we take some time at the end to discuss the difference between the two frequencies. Whose spinner was the most accurate? How do we know this? What factors could have influenced the spinner landing on one category more than expected? How could we reduce bias in future experiments? Could the sample size be altered to make the findings more representative?

What’s next?
The second lesson goes on to use the experimental probabilities to create odds for a gambling game. More on this next week…

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