L01.2 - Sample Space

Table of Contents:

A) Review: Probabilistic Models

Putting together a probabilistic model (a model of a random phenomenon or experiment) involves two key steps:

1st Step) "Sample Space", it describes possible outcomes of the phenomenon or experiment of interest

2nd Step) Describe our beliefs about the likelihood of the different possible outcomes by specifying a probability law

B) Step1: The Sample Space

Let's now study the first step in detail.

For example: We carry out an experiment, we can:

flip a coin
flip 5 coins simultaneously
roll a die

Whatever that experiment is, it has a number of possible outcomes.

B.1) Sample Space "Set" Requirements

We start by making a list (better known as a "Set") of ALL possible outcomes of our experiment. The set of the sample space is denoted by $Ω$

The elements of set $Ω$ should have certain properties, the sample space should be:

B.1.1) Mutually Exclusive

① Mutually Exclusive: it means that at the end of the experiment only ONE of the possible outcomes will happen. There is no overlap.

B.1.2) Collectively Exhaustive

② Collectively exhaustive: it means that TOGETHER all the element of the set "exhaust" ALL the possibilities.

In other words, we know for sure that at the end of the experiment, the end result is one element of the sample space

B.1.3) "Right" Granularity

③ At the "right"granularity: it means the elements of the set contain enough information to answer questions about the experiment.
(The model has the "right" level of detail)

For example: Consider the experiment of flipping a coin.

We could have 2 sample spaces:

Both are "legitimate" sample spaces for the experiment of flipping a coin.

The element are:
① Mutually Exclusive
② Collectively exhaustive

So, which sample space should we use?

We could argue that the second sample space involves some irrelevant details.

So the preferred sample space for describing flipping of a coin is the first one, we say it is at the "right"granularity given the questions we want to answer.

But if you have a "theory" on how the weather affects the flipping of coins you could use the second sample space.

In Science, wherever you put together a model, you need to decide how detailed you want your model to be. And the right level of detail is the one that captures the aspects that are relevant and of interest to you.

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