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Suppose I look at the registry
of residents of my town and

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pick a person at random.

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What is the probability
that this person is

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under 18 years of age?

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The answer is about 25%.

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Suppose now that I tell you that
this person is married.

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Will you give the same answer?

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Of course not.

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The probability of being
less than 18 years

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old is now much smaller.

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What happened here?

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We started with some initial
probabilities that reflect

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what we know or believe
about the world.

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But we then acquired
some additional

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knowledge, some new evidence--

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for example, about this person's
family situation.

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This new knowledge should cause
our beliefs to change,

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and the original probabilities
must be replaced with new

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probabilities that take into
account the new information.

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These revised probabilities are
what we call conditional

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probabilities.

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And this is the subject
of this lecture.

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We will start with a formal
definition of conditional

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probabilities together with
the motivation behind this

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particular definition.

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We will then proceed to develop
three tools that rely

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on conditional probabilities,
including the Bayes rule,

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which provides a systematic
way for incorporating new

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evidence into a probability
model.

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The three tools that we
introduce in this lecture

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involve very simple and
elementary mathematical

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formulas, yet they encapsulate
some very powerful ideas.

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It is not an exaggeration to
say that much of this class

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will revolve around the repeated
application of

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variations of these three
tools to increasingly

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complicated situations.

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In particular, the Bayes rule
is the foundation for the

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field of inference.

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It is a guide on how to
process data and make

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inferences about unobserved
quantities or phenomena.

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As such, it is a tool that is
used all the time, all over

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science and engineering.