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Let us now look at some examples
of sample spaces.

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Sample spaces are sets.

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And a set can be discrete,
finite, infinite,

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continuous, and so on.

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Let us start with a simpler case
in which we have a sample

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space that is discrete
and finite.

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The particular experiment
we will be

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looking at is the following.

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We take a very special die,
a tetrahedral die.

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So it's a die that has four
faces numbered from 1 up 4.

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We roll it once.

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And then we roll it
twice [again].

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Were not dealing here with two
probabilistic experiments.

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We're dealing with a single
probabilistic experiment that

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involves two rolls of the die
within that experiment.

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What is the sample space
of that experiment?

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Well, one possible

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representation is the following.

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We take note of the result
of the first roll.

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And then we take note of the
result of the second roll.

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And this gives us a
pair of numbers.

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Each one of the possible pairs
of numbers corresponds to one

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of the little squares
in this diagram.

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For example, if the first roll
is 1 and the second is also 1,

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then this particular outcome
has occurred.

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If the first roll is it 2 and
the second is a 3, then this

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particular outcome occurs.

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If the first roll is a 3 and
then the next one is a 2, then

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this particular outcome
occurs.

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Notice that these two outcomes
are pretty closely related.

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In both cases, we observe
a 2 and we observe a 3.

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But we distinguish those two
outcomes because in those two

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outcomes, the 2 and the 3 happen
in different order.

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And the order in which they
appear may be a detail which

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is of interest to us.

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And so we make this distinction

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in the sample space.

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So we keep the (3, 2) and the
(2, 3) as separate outcomes.

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Now this is a case of a model
in which the probabilistic

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experiment can be described
in phases or stages.

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We could think about rolling
the die once and then going

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ahead with the second roll.

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So we have two stages.

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A very useful way of describing
the sample space of

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experiments--

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whenever we have an experiment
with several stages, either

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real stages or imagined
stages.

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So a very useful way of
describing it is by providing

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a sequential description
in terms of a tree.

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So a diagram of this kind,
we call it a tree.

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You can think of this as
the root of the tree

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from which you start.

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And the endpoints of the
tree, we usually

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call them the leaves.

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So the experiment starts.

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We carry out the first phase,
which in this case is the

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first roll.

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And we see what happens.

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So maybe we get a 2
in the first roll.

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And then we take note of what
happened in the second roll.

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And maybe the result was a 3.

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So we follow this branch here.

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And we end up at this particular
leaf, which is the

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leaf associated with
the outcome 2, 3.

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Notice that in this tree we once
more have a distinction.

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The outcome 2 followed by a 3 is
different from the outcome

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3 followed by a 2, which would
correspond to this particular

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place in the diagram.

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In both cases, we have
16 possible outcomes.

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4 times 4 makes 16.

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And similarly, if you count
here, the number of leaves is

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equal to 16.

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The previous example involves
a sample space that was

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discrete and finite.

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There were only 16 possible
outcomes.

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But sample spaces can
also be infinite.

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And they could also be
continuous sets.

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Here's an example of an
experiment that involves a

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continuous sample space.

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So we have a rectangular target

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which is the unit square.

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And you throw a dart
on that target.

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And suppose that you are so
skilled that no matter what,

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when you throw the
dart, it always

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falls inside the target.

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Once the dart hits the target,
you record the coordinates x

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and y of the particular
point that resulted

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from your dart throw.

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And we record x and y with
infinite precision.

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So x and y are real numbers.

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So in this experiment, the
sample space is just the set

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of x, y pairs that lie
between 0 and 1.