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We end this lecture sequence
by stepping back to discuss

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what probability theory really
is and what exactly is the

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meaning of the word
probability.

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In the most narrow view,
probability theory is just a

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branch of mathematics.

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We start with some axioms.

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We consider models that satisfy
these axioms, and we

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establish some consequences,
which are the

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theorems of this theory.

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You could do all that without
ever asking the question of

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what the word "probability"
really means.

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Yet, one of the theorems of
probability theory, that we

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will see later in this class,
is that probabilities can be

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interpreted as frequencies,
very loosely speaking.

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If I have a fair coin, and I
toss it infinitely many times,

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then the fraction of
heads that I will

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observe will be one half.

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In this sense, the probability
of an event, A, can be

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interpreted as the frequency
with which event A will occur

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in an infinite number of
repetitions of the experiment.

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But is this all there is?

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If we're dealing with coin
tosses, it makes sense to

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think of probabilities
as frequencies.

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But consider a statement such
as the "current president of

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my country will be reelected
in the next election with

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probability 0.7".

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It's hard to think of this
number, 0.7, as a frequency.

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It does not make sense to
think of infinitely many

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repetitions of the
next election.

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In cases like this, and in many
others, it is better to

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think of probabilities
as just some way of

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describing our beliefs.

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And if you're a betting person,
probabilities can be

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thought of as some numerical
guidance into what kinds of

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bets you might be
willing to make.

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But now if we think of
probabilities as beliefs, you

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can run into the argument
that, well, beliefs are

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subjective.

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Isn't probability theory
supposed to be an objective

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part of math and science?

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Is probability theory just an
exercise in subjectivity?

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Well, not quite.

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There's more to it.

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Probability, at the minimum,
gives us some rules for

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thinking systematically about
uncertain situations.

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And if it happens that our
probability model, our

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subjective beliefs, have some
relation with the real world,

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then probability theory can be
a very useful tool for making

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predictions and decisions that
apply to the real world.

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Now, whether your predictions
and decisions will be any good

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will depend on whether you
have chosen a good model.

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Have you chosen a model that's
provides a good enough

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representation of
the real world?

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How do you make sure that
this is the case?

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There's a whole field, the field
of statistics, whose

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purpose is to complement
probability theory by using

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data to come up with
good models.

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And so we have the following
diagram that summarizes the

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relation between the real
world, statistics, and

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probability.

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The real world generates data.

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The field of statistics and
inference uses these data to

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come up with probabilistic
models.

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Once we have a probabilistic
model, we use probability

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theory and the analysis tools
that it provides to us.

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And the results that we get
from this analysis lead to

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predictions and decisions
about the real world.