Lesson 1. Sample space

Putting together a probabilistic Model that is, a model of a random
phenomenon or a random experiment involves two steps.

First step, we describe the possible outcomes of the phenomenon or experiment
of interest.

Second step, we describe our beliefs about the likelihood of the different possible outcomes by specifying a probability law.

Here, we start by just talking about the first step, namely, the description of the possible outcomes of the experiment.

So we carry out an experiment. For example, we flip a coin. Or maybe we flip five coins simultaneously.

Or maybe we roll a die.

Whatever that experiment is,
it has a number of possible

outcomes, and we start
by making a list of

the possible outcomes--

or, a better word, instead of
the word "list", is to use the

word "set", which has a more
formal mathematical meaning.

So we create a set
that we usually

denote by capital omega.

That set is called the sample
space and is the set of all

possible outcomes of
our experiment.

The elements of that set
should have certain

properties.

Namely, the elements should
be mutually exclusive and

collectively exhaustive.

What does that mean?

Mutually exclusive means that,
if at the end of the

experiment, I tell you that this
outcome happened, then it
should not be possible that this
outcome also happened.

At the end of the experiment,
there can only be one of the

outcomes that has happened.

Being collectively exhaustive
means something else-- that,

together, all of these elements
of the set exhaust all the possibilities.

So no matter what, at the end,
you will be able to point to

one of the outcomes and say,
that's the one that occurred.

To summarize--

this set should be such that, at
the end of the experiment,

you should be always able to
point to one, and exactly one,

of the possible outcomes and say
that this is the outcome
that occurred.

Physically different outcomes
should be distinguished in the

sample space and correspond
to distinct points.


But when we say physically
different

outcomes, what do we mean?

We really mean different in
all relevant aspects but

perhaps not different in
irrelevant aspects.

Let's make more precise what I
mean by that by looking at a

very simple, and maybe
silly, example,

which is the following.

Suppose that you flip a coin
and you see whether it

resulted in heads or tails.

So you have a perfectly
legitimate sample space for

this experiment which consists
of just two points--

heads and tails.
Together these two outcomes
exhaust all possibilities.

And the two outcomes are
mutually exclusive.
So this is a very legitimate
sample space for this

experiment.

Now suppose that while you were
flipping the coin, you

also looked outside the window
to check the weather.

And then you could say that my
sample space is really, heads,

and it's raining.

Another possible outcome
is heads and no rain.

Another possible outcome is
tails, and it's raining, and,

finally, another possible
outcome is tails and no rain.


This set, consisting of four
elements, is also a perfectly

legitimate sample space
for the experiment
of flipping a coin.

The elements of this sample
space are mutually exclusive

and collectively exhaustive.

Exactly one of these outcomes
is going to be true, or will

have materialized, at the
end of the experiment.

So which sample space
is the correct one?

This sample space, the
second one, involves

some irrelevant details.

So the preferred sample space
for describing the flipping of

a coin, the preferred sample
space is the simpler one, the

first one, which is sort of at
the right granularity, given

what we're interested in.

But ultimately, the question
of which one is the right

sample space depends
on what kind of

questions you want to answer.

For example, if you have a
theory that the weather

affects the behavior of coins,
then, in order to play with

that theory, or maybe check it
out, and so on, then, in such

a case, you might want to work
with the second sample space.

This is a common feature
in all of science.

Whenever 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 those aspects that are relevant and of interest to you.