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In the previous example, we had
a model where the result

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of the first coin toss did not
affect the probabilities of

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what might happen in
the second toss.

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This is a phenomenon that we
call independence and which we

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now proceed to define.

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Let us start with a first
attempt at the definition.

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We have an event, B,
that has a certain

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probability of occurring.

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We are then told that event A
occurred, but suppose that

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this knowledge does not affect
our beliefs about B in the

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sense that the conditional
probability remains the same

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as the original unconditional
probability.

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Thus, the occurrence of A
provides no new information

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about B. In such a case, we
may say that event B is

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independent from event A.

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If this is indeed the case,
notice that the probability

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that both A and B occur, which
is always equal by the

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multiplication rule to the
probability of A times the

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conditional probability of B
given A. So this is a relation

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that's always true.

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But if we also have this
additional condition, then

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this simplifies to the
probability of A times the

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probability of B.

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So we can find the probability
of both events happening by

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just multiplying their
individual probabilities.

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It turns out that this relation
is a cleaner way of

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the defining formally the
notion of independence.

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So we will say that two
events, A and B, are

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independent if this
relation holds.

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Why do we use this definition
rather than the original one?

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This formal definition has
several advantages.

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First, it is consistent with
the earlier definition.

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If this equality is true, then
the conditional probability of

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event B given A, which is the
ratio of this divided by that,

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will be equal to the probability
of B. So if this

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relation holds, then this
relation will also hold, and

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so this more formal definition
is consistent with our earlier

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intuitive definition.

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A more important reason is that
this formal definition is

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symmetric with respect to the
roles of A and B. So instead

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of saying that B is independent
from A, based on

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this definition we can now say
that events A and B are

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independent of each other.

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And in addition, since this
definition is symmetric and

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since it implies this condition,
it must also imply

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the symmetrical relation.

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Namely, that the conditional
probability of A given B is

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the same as the unconditional
probability of A.

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Finally, on the technical
side, conditional

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probabilities are only defined
when the conditioning event

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has non-zero probability.

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So this original definition
would only make sense in those

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cases where the probability of
the event A would be non-zero.

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In contrast, this new definition
makes sense even

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when we're dealing with zero
probability events.

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So this definition is indeed
more general, and this also

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makes it more elegant.

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Let us now build some
understanding of what

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independence really is.

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Suppose that we have two events,
A and B, both of which

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have positive probability.

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And furthermore, these two
events are disjoint.

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They do not have any
common elements.

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Are these two events
independent?

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Let us check the definition.

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The probability that both A and
B occur is zero because

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the two events are disjoint.

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They cannot happen together.

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On the other hand, the
probability of A times the

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probability of B is positive,
since each one of the two

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terms is positive.

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And therefore, these two
expressions are different from

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each other, and therefore this
equality that's required by

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the definition of independence
does not hold.

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The conclusion is that these
two events are not

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independent.

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In fact, intuitively, these two
events are as dependent as

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Siamese twins.

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If you know that A occurred,
then you are sure

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that B did not occur.

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So the occurrence of A tells you
a lot about the occurrence

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or non-occurrence of B.

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So we see that being independent
is something

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completely different from
being disjoint.

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Independence is a relation
about information.

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It is important to always keep
in mind the intuitive meaning

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of independence.

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Two events are independent if
the occurrence of one event

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does not change our beliefs
about the other.

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It does not affect the
probability that the other

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event also occurs.

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When do we have independence
in the real world?

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The typical case is when the
occurrence or non-occurrence

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of each of the two events A
and B is determined by two

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physically distinct and
non-interacting processes.

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For example, whether my coin
results in heads and whether

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it will be snowing on New Year's
Day are two events that

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should be modeled
as independent.

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But I should also say that there
are some cases where

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independence is less obvious and
where it happens through a

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numerical accident.

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You can now move on to answer
some simple questions where

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you will have to check for
independence using either the

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mathematical or intuitive
definition.