Inferring the Fraction of Blue M&Ms, Part 1: Problem Set-Up

Overview

Teaching: 0 min
Exercises: 0 min

Questions

• What statistical questions can we ask about M&Ms?

Objectives

• First learning objective. (FIXME)

Now that we’ve defined Bayes’ theorem in the previous episode, we’re going to use it for a real-world problem. Since all that talk of snacks during the episode on conditional probabilities has made me hungry, we’re going to talk about M&Ms.

M&Ms are chocolate candies that come in bags, where each bag has some number of different M&Ms. They come in different types (for example, some have peanuts in them, some have almonds, and some no nuts at all), and in different colours.

Collecting Questions

Given a bag of M&M candies, what kind of information can you learn from the M&Ms in that bag? Can you think of a question you might be able to answer by exploring the contents of your bag?

Write down an idea, then share and discuss with your neighbour. What data would you need to collect from your bag of M&Ms to answer your question?

Note: Don’t open the bag yet! We’ll get to that in a little while.

There are a number of different questions you might be able to answer with a bag of M&Ms.

Here’s the basic problem we’re going to talk about today: You have a bag of M&Ms, and you want to know how many blue ones are in it. Easy, right? You open the bag, and count all the blue ones! But what if you have a 10 pound bag of M&Ms? Counting them all would take you ages, wouldn’t it? Or what if you had several bags. The exact number of blue M&Ms as well as the total number of M&Ms in each bag might vary! So really, what we need is a model of the average number of blue M&Ms per bag. We don’t just need any model, but we need a statistical model that describes the number of blue M&Ms we would get out of a bag of M&Ms given some underlying true fraction of blue M&M that were produced at the factory.

So, in short, our research question for today is this:

Our Research Question

What is the percentage of blue M&Ms produced at the factory?

To answer this question, we will be using Bayes’ theorem and one bag of M&Ms. Here, our data $D$ will be the M&Ms in our bag, along with their colour (for simplicity, we will use two colours: “blue” and “not-blue”). The parameter $\theta$ we are interested in here is the percentage of blue M&Ms produced at the factory.

We will need to define some additional details about the problem, particularly the likelihood and the prior, but let’s do a little exercise first:

Some Initial Questions

1. In the data visualization you have learned that there are different types of data. What type of data, do you think, are M&Ms?
   • continuous
   • ordinal
   • categorical

If you don’t remember what those words mean, you can take a look at the brief discussion about types of data in the visualization tutorial.

1. How will you record the data?
2. Will you draw M&Ms out of the bag with replacement or without?
3. How might we model the probability of drawing a blue m&m?

Solution

1. Our data is categorical data: we have two different colours: blue and not-blue, and we are measuring the probability of obtaining each category when we draw an m&m out of the bag.
2. We will record the data in a webform so that we can then continue to use it.
3. We will be sampling without replacement, that is, once we draw an M&M out of the bag, we won’t put it back inside.

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Key Points

• First key point. Brief Answer to questions. (FIXME)