#edx-probability

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.

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.

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