WEBVTT
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This is a simple example where
we want to just apply the
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formula for conditional
probabilities
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and see what we get.
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The example involves a
four-sided die, if you can
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imagine such an object, which
we roll twice, and we record
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the first roll, and
the second roll.
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So there are 16 possible
outcomes.
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We assume to keep things simple,
that each one of those
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16 possible outcomes, each
one of them has the same
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probability, so each outcome
has the probability 1/16.
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Let us consider now a particular
event B on which
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we're going to condition.
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This is the event under which
the smaller of the two die
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rolls is equal to 2, which means
that one of the dice
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must have resulted in two, and
the other die has resulted in
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something which is
2 or larger.
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So this can happen
in multiple ways.
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And here are the different
ways that it can happen.
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So at 2, 2, or 2, 3, or 2, 4;
then a 3, 2 and a 4, 2.
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All of these are outcomes in
which one of the dice has a
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value equal to 2, and
the other die
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is at least as large.
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So we condition on this event.
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This results in a conditional
model where each one of those
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five outcomes are equally likely
since they used to be
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equally likely in the
original model.
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Now let's look at
this quantity.
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The maximum of the
two die rolls--
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that is, the largest
of the results.
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And let us try to calculate
the following quantity--
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the conditional probability that
the maximum is equal to 1
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given that the minimum
is equal to 2.
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So this is the conditional
probability of
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this particular outcome.
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Well, this particular outcome
cannot happen.
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If I tell you that the smaller
number is 2, then the larger
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number cannot be equal to 1, so
this outcome is impossible,
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and therefore this conditional
probability is equal to 0.
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Let's do something a little
more interesting.
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Let us now look at the
conditional probability that
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the maximum is equal to 3 given
the information that
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event B has occurred.
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It's best to draw a picture
and see what that event
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corresponds to.
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M is equal to 3--
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the maximum is equal to 3--
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if one of the dice resulted
in a 3, and the other die
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resulted in something
that's 3 or less.
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So this event here corresponds
to the blue
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region in this diagram.
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Now let us try to calculate the
conditional probability by
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just following the definition.
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The conditional probability of
one event given another is the
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probability that
both of them--
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both of the two events--
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occur, divided by the
probability of the
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conditioning event.
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That is, out of the total
probability in the
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conditioning event, we ask,
what fraction of that
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probability is assigned to
outcomes in which the event of
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interest is also happening?
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So what is this event?
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The maximum is equal to 3,
which is the blue event.
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And simultaneously, the red
event is happening.
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These two events intersect
only in two places.
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This is the intersection
of the two events.
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And the probability of that
intersection is 2 out of 16,
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since there's 16 outcomes and
that event happens only with
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two particular outcomes.
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So this gives us 2/16
in the numerator.
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How about the denominator?
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Event B consists of a total
of five possible outcomes.
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Each one has probability 1/16,
so this is 5/16, so the final
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answer is 2/5.
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We could have gotten that same
answer in a simple and perhaps
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more intuitive way.
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In the original model, all
outcomes were equally likely.
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Therefore, in the conditional
model, the five outcomes that
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belong to B should also
be equally likely.
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Out of those five, there's two
of them that make the event of
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interest to occur.
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So given that we live in B,
there's two ways out of five
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that the event of interest
will materialize.
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So the event of interest has
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conditional probability [equal to]
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2/5.