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PROFESSOR: Independent events
are events that have nothing
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to do with each other.
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And needless to say, it's a
lot easier to work with them
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because you don't
have to figure out
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some weird interaction
between two events that
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do have something to
do with each other.
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Typical case where
independent events come up
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is, for example, you
toss a coin five times,
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and then you're about to
toss a coin the sixth time.
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Is there any reason
to think that
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what the coins did the
first five times is going
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to have any
influence on the flip
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of the coin for the sixth time?
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Well, it's reasonable
to assume not,
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which is to say that the
outcome of the sixth toss
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is arguably independent
of the outcome of all
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the previous tosses.
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OK.
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Let's look at a
formal definition
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now in this short
video of just what
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is the technical definition
of independent events.
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So what we said is that they
are independent if they have
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nothing to do with each other.
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What that means is that if
I tell you that B happened,
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it doesn't have any effect on
the probability of A. That is,
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the probability of A,
given that B happened,
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doesn't change the
probability of A at all.
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That's it.
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Now this is one definition.
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Maybe this is the more
intuitive definition.
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But another definition that's
equivalent and is standard
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is that two events are
equivalent if the product
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of their probabilities is
equal to the probability
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that they both happen,
that is, the probability
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of their intersection.
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Now definition one
and definition two
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are trivial equivalent,
just using the definition
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of conditional probability.
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And if you don't see that,
this would be a moment
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to stop, get a pencil
and paper, and write down
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the three-line proof of the
equivalence of these two
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equalities.
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In fact, the three-line proof
has this as the first line
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and that as the second line.
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So you could argue it's
really just the middle line
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that you're adding.
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OK.
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Definition two has the slight
advantage that it always works,
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whereas definition
one implicitly
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requires that the divisor--
remember probability of A given
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B is defined as the probability
of the intersection divided
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by the probability B. It's only
defined if the probability of B
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is positive.
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Whereas the second
definition always works,
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so we don't have
to put a proviso
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in about the probability
of B being non-zero.
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So that's the definition
of independence.
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Looking at this definition,
what you can see immediately
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is that it's completely
symmetric in A and B.
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Since multiplication
is commutative
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and intersection is
commutative, which is A
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and which is B doesn't matter.
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And what that implies then
is that A is independent of B
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if and only if B is
independent of A.
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Now another fact that holds is
that if the probability of B
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happens to be zero,
then vacuously B
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is independent of
everything-- even itself.
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Which isn't important, but is a
small technicality that's worth
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remembering by that definition.
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Now again, the intuitive idea
that A and B have nothing
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to do with each other is that
A is independent of B means
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that A is independent of
whether or not B occurs.
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That is to say, if A
is independent of B,
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it ought to be independent
of the complement of B.
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And that's a lemma that's
also easily proved.
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A is independent
of B if and only
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if A is independent of
the complement of B.
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It's a simple proof using
the difference rule.
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And again, I
encourage you to stop
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with a piece of
paper and a pencil
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and convince yourself that that
follows with a one-line proof.